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#### Question 1:

Compute the following sums:
(i)

(ii)

#### Question 2:

Let A = $\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right]$, B = and C = . Find each of the following:
(i) 2A − 3B
(ii) B − 4C
(iii) 3AC
(iv) 3A − 2B + 3C

#### Question 3:

If A = $\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right]$, B = , C = , find
(i) A + B and B + C
(ii) 2B + 3A and 3C − 4B.

It is not possible to add these matrices because the number of elements in A are not equal to the
number of elements in B. So, A + B does not exist.

It is not possible to add these matrices because the number of elements in B are not equal to the
number of elements in A. So, 2B + 3A does not exist.

$\phantom{\rule{0ex}{0ex}}⇒3C-4B=3\left[\begin{array}{ccc}-1& 2& 3\\ 2& 1& 0\end{array}\right]-4\left[\begin{array}{ccc}-1& 0& 2\\ 3& 4& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒3C-4B=\left[\begin{array}{ccc}-3& 6& 9\\ 6& 3& 0\end{array}\right]-\left[\begin{array}{ccc}-4& 0& 8\\ 12& 16& 4\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒3C-4B=\left[\begin{array}{ccc}-3+4& 6-0& 9-8\\ 6-12& 3-16& 0-4\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒3C-4B=\left[\begin{array}{ccc}1& 6& 1\\ -6& -13& -4\end{array}\right]$

#### Question 4:

Let A = B = $\left[\begin{array}{ccc}0& -2& 5\\ 1& -3& 1\end{array}\right]$ and C = . Compute 2A − 3B + 4C.

$\mathrm{Here},\phantom{\rule{0ex}{0ex}}2A-3B+4C=2\left[\begin{array}{ccc}-1& 0& 2\\ 3& 1& 4\end{array}\right]-3\left[\begin{array}{ccc}0& -2& 5\\ 1& -3& 1\end{array}\right]+4\left[\begin{array}{ccc}1& -5& 2\\ 6& 0& -4\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒2A-3B+4C=\left[\begin{array}{ccc}-2& 0& 4\\ 6& 2& 8\end{array}\right]-\left[\begin{array}{ccc}0& -6& 15\\ 3& -9& 3\end{array}\right]+\left[\begin{array}{ccc}4& -20& 8\\ 24& 0& -16\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒2A-3B+4C=\left[\begin{array}{ccc}-2-0+4& 0+6-20& 4-15+8\\ 6-3+24& 2+9+0& 8-3-16\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒2A-3B+4C=\left[\begin{array}{ccc}2& -14& -3\\ 27& 11& -11\end{array}\right]$

#### Question 5:

If A = diag (2 − 59), B = diag (11 − 4) and C = diag (−6 3 4), find
(i) A − 2B
(ii) B + C − 2A
(iii) 2A + 3B − 5C

#### Question 6:

Given the matrices
A = , B = and C =
Verify that (A + B) + C = A + (B + C).

Hence proved.

#### Question 7:

Find matrices X and Y, if X + Y = $\left[\begin{array}{cc}5& 2\\ 0& 9\end{array}\right]$ and XY =

#### Question 8:

Find X if Y = $\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right]$ and 2X + Y =

$\mathrm{Given}: 2X+Y=\left[\begin{array}{cc}1& 0\\ -3& 2\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒2X+\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right]=\left[\begin{array}{cc}1& 0\\ -3& 2\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒2X=\left[\begin{array}{cc}1& 0\\ -3& 2\end{array}\right]-\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒2X=\left[\begin{array}{cc}1-3& 0-2\\ -3-1& 2-4\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒2X=\left[\begin{array}{cc}-2& -2\\ -4& -2\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒X=\frac{1}{2}\left[\begin{array}{cc}-2& -2\\ -4& -2\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒X=\left[\begin{array}{cc}-1& -1\\ -2& -1\end{array}\right]$

#### Question 9:

Find matrices X and Y, if 2XY = and X + 2Y =

#### Question 10:

If XY = $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 0\\ 1& 0& 0\end{array}\right]$ and X + Y = , find X and Y.

#### Question 11:

Find matrix A, if + A =

$\mathrm{Here},\phantom{\rule{0ex}{0ex}}A=\left[\begin{array}{ccc}9& -1& 4\\ -2& 1& 3\end{array}\right]-\left[\begin{array}{ccc}1& 2& -1\\ 0& 4& 9\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒A=\left[\begin{array}{ccc}9-1& -1-2& 4+1\\ -2-0& 1-4& 3-9\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒A=\left[\begin{array}{ccc}8& -3& 5\\ -2& -3& -6\end{array}\right]$

#### Question 12:

If A = $\left[\begin{array}{cc}9& 1\\ 7& 8\end{array}\right]$, B = $\left[\begin{array}{cc}1& 5\\ 7& 12\end{array}\right]$, find matrix C such that 5A + 3B + 2C is a null matrix.

#### Question 13:

If A = , B = , find matrix X such that 2A + 3X = 5B.

#### Question 14:

If A = and, B = , find the matrix C such that A + B + C is zero matrix.

$\mathrm{Given}: A+B+C=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒\left[\begin{array}{ccc}1& -3& 2\\ 2& 0& 2\end{array}\right]+\left[\begin{array}{ccc}2& -1& -1\\ 1& 0& -1\end{array}\right]+C=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒\left[\begin{array}{ccc}1+2& -3-1& 2-1\\ 2+1& 0+0& 2-1\end{array}\right]+C=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒\left[\begin{array}{ccc}3& -4& 1\\ 3& 0& 1\end{array}\right]+C=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒C=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]-\left[\begin{array}{ccc}3& -4& 1\\ 3& 0& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒C=\left[\begin{array}{ccc}0-3& 0+4& 0-1\\ 0-3& 0-0& 0-1\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒C=\left[\begin{array}{ccc}-3& 4& -1\\ -3& 0& -1\end{array}\right]$

#### Question 15:

Find x, y satisfying the matrix equations

(i)

(ii)

(iii) $x\left[\begin{array}{c}2\\ 1\end{array}\right]+y\left[\begin{array}{c}3\\ 5\end{array}\right]+\left[\begin{array}{c}-8\\ -11\end{array}\right]=0$

#### Question 16:

If 2, find x and y.

#### Question 17:

Find the value of λ, a non-zero scalar, if λ

#### Question 18:

(i) Find a matrix X such that 2A + B + X = O, where
A = , B =
(ii) If A = and B = , then find the matrix X of order 3 × 2 such that 2A + 3X = 5B.

#### Question 19:

Find x, y, z and t, if
(i)

(ii)

#### Question 20:

If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.

We have,

Also,

From (1) and (2), we get

.

#### Question 21:

In a certain city there are 30 colleges. Each college has 15 peons, 6 clerks, 1 typist and 1 section officer. Express the given information as a column matrix. Using scalar multiplication, find the total number of posts of each kind in all the colleges.

Number of different types of posts in any college is given by

X  = $\left[\begin{array}{c}15\\ 6\\ 1\\ 1\end{array}\right]$

Total number of posts of each kind in all the colleges = 30X

= 30$\left[\begin{array}{c}15\\ 6\\ 1\\ 1\end{array}\right]$

=  $\left[\begin{array}{c}450\\ 180\\ 30\\ 30\end{array}\right]$

#### Question 22:

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

Let the monthly incomes of Aryan and Babban be 3x and 4x, respectively.

Suppose their monthly expenditures are 5y and 7y, respectively.

Since each saves Rs 15,000 per month,

The above system of equations can be written in the matrix form as follows:

$\left[\begin{array}{cc}3& -5\\ 4& -7\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}15000\\ 15000\end{array}\right]$

or,
AX = B, where

Now,

$\left|\mathrm{A}\right|=\left|\begin{array}{cc}3& -5\\ 4& -7\end{array}\right|=-21-\left(-20\right)=-1$

Adj A=${\left[\begin{array}{cc}-7& -4\\ 5& 3\end{array}\right]}^{T}=\left[\begin{array}{cc}-7& 5\\ -4& 3\end{array}\right]$

So, ${A}^{-1}=\frac{1}{\left|A\right|}adjA=-1\left[\begin{array}{cc}-7& 5\\ -4& 3\end{array}\right]=\left[\begin{array}{cc}7& -5\\ 4& -3\end{array}\right]$

Therefore,

Monthly income of Aryan =

Monthly income of Babban =

From this problem, we are encouraged to understand the power of savings. We should save certain part of our monthly income for the future.

#### Question 1:

Compute the indicated products:
(i)

(ii)

(iii)

#### Question 2:

Show that ABBA in each of the following cases:
(i)

(ii)

(iii)

#### Question 3:

Compute the products AB and BA whichever exists in each of the following cases:
(i)

(ii)
(iii) A = [1 −1 2 3] and $B=\left[\begin{array}{c}0\\ 1\\ 3\\ 2\end{array}\right]$

(iv) [a, b]$\left[\begin{array}{c}c\\ d\end{array}\right]$ + [a, b, c, d]$\left[\begin{array}{c}a\\ b\\ c\\ d\end{array}\right]$

Since the number of columns in B is greater then the number of rows in A, BA does not exists.

#### Question 4:

Show that ABBA in each of the following cases:
(i)

(ii)

#### Question 5:

Evaluate the following:
(i)

(ii)

(iii)

#### Question 6:

If A = $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, B = and C = $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, then show that A2 = B2 = C2 = I2.

#### Question 7:

If A = and B = , find 3A2 − 2B + I

#### Question 8:

If A = , prove that (A − 2I) (A − 3I) = O

#### Question 9:

If A = $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$, show that A2 = $\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]$ and A3 = $\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]$.

Hence proved.

#### Question 10:

If A = , show that A2 = O

#### Question 11:

If A = , find A2.

#### Question 12:

If A = and B = , show that AB = BA = O3×3.

#### Question 13:

If A = and B = $\left[\begin{array}{ccc}{a}^{2}& ab& ac\\ ab& {b}^{2}& bc\\ ac& bc& {c}^{2}\end{array}\right]$, show that AB = BA = O3×3.

#### Question 14:

If A = and B = , show that AB = A and BA = B.

#### Question 15:

Let A = and B = , compute A2B2.

#### Question 16:

For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A (BC):
(i)

(ii) .

#### Question 17:

For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC:
(i)
(ii)

#### Question 18:

If , verify that A (BC) = ABAC.

#### Question 19:

Compute the elements a43 and a22 of the matrix:

We have,

#### Question 20:

If $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ p& q& r\end{array}\right]$, and I is the identity matrix of order 3, show that A3 = pI + qA +rA2.

#### Question 21:

If w is a complex cube root of unity, show that

#### Question 22:

If , show that A2 = A.

#### Question 23:

If , show that A2 = I3.

#### Question 24:

(i) If [1 1 x]$\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 2& 1& 0\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$ = 0, find x.
(ii) If $\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right]\left[\begin{array}{cc}1& -3\\ -2& 4\end{array}\right]=\left[\begin{array}{cc}-4& 6\\ -9& x\end{array}\right]$ , find x.

(i)

(ii)

#### Question 25:

If [x 4 1] = 0, find x.

#### Question 26:

If [1 −1 x] = 0, find x.

#### Question 27:

If , then prove that A2A + 2I = O.

#### Question 28:

If , then find λ so that A2 = 5A + λI.

#### Question 29:

If , show that A2 − 5A + 7I2 = O

#### Question 30:

If , show that A2 − 2A + 3I2 = O

#### Question 31:

Show that the matrix $A=\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right]$ satisfies the equation A3 − 4A2 + A = O

#### Question 32:

Show that the matrix is root of the equation A2 − 12AI = O

#### Question 33:

If , find A2 − 3A − 7I.

#### Question 34:

If , show that A2 − 5A + 7I = O use this to find A4.

#### Question 35:

If $A=\left[\begin{array}{cc}3& -2\\ 4& -2\end{array}\right]$, find k such that A2 = kA − 2I2

#### Question 36:

If , find k such that A2 − 8A + kI = 0.

#### Question 37:

If $A=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$, f (x) = x2 − 2x − 3, show that f (A) = 0

#### Question 38:

If  then find λ, μ so that A2 = λA + μI

#### Question 39:

Find the value of x for which the matrix product
equal an identity matrix.

#### Question 40:

Solve the matrix equations:
(i)

(ii)

(iii) $\left[x-5-1\right]\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 2& 0& 3\end{array}\right]\left[\begin{array}{c}x\\ 4\\ 1\end{array}\right]=0$

(iv) $\left[\begin{array}{cc}2x& 3\end{array}\right]\left[\begin{array}{cc}1& 2\\ -3& 0\end{array}\right]\left[\begin{array}{c}x\\ 8\end{array}\right]=0$

#### Question 41:

If , compute A2 − 4A + 3I3.

#### Question 42:

If f (x) = x2 − 2x, find f (A), where $A=\left[\begin{array}{ccc}0& 1& 2\\ 4& 5& 0\\ 0& 2& 3\end{array}\right]$

#### Question 43:

If f (x) = x3 + 4x2x, find f (A), where

#### Question 44:

If $A=\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 2& 0& 3\end{array}\right]$, then show that A is a root of the polynomial f (x) = x3 − 6x2 + 7x + 2.

#### Question 45:

If $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$, then prove that A2 − 4A − 5I = O.

Hence proved.

#### Question 46:

If $A=\left[\begin{array}{ccc}3& 2& 0\\ 1& 4& 0\\ 0& 0& 5\end{array}\right]$, show that A2 − 7A + 10I3 = O

#### Question 47:

Without using the concept of inverse of a matrix, find the matrix $\left[\begin{array}{cc}x& y\\ z& u\end{array}\right]$ such that

#### Question 48:

Find the matrix A such that
(i)

(ii)

(iii)

(iv) $\left[\begin{array}{ccc}2& 1& 3\end{array}\right]\left[\begin{array}{ccc}-1& 0& -1\\ -1& 1& 0\\ 0& 1& 1\end{array}\right]\left[\begin{array}{c}1\\ 0\\ -1\end{array}\right]=A$

(v)   A

(vi) A

#### Question 49:

Find a 2 × 2 matrix A such that

Let A = $\left[\begin{array}{cc}w& x\\ y& z\end{array}\right]$

Now,

#### Question 50:

If $A=\left[\begin{array}{cc}0& 0\\ 4& 0\end{array}\right]$, find A16.

#### Question 51:

If  and x2 = −1, then show that (A + B)2 = A2 + B2.

Given:  and x2 = −1

To show: (A + B)2 = A2 + B2

LHS:

RHS:

Comparing (1) and (4), we get

(A + B)2 = A2 + B2

#### Question 52:

If $A=\left[\begin{array}{ccc}1& 0& -3\\ 2& 1& 3\\ 0& 1& 1\end{array}\right]$, then verify that A2 + A = A(A + I), where I is the identity matrix.

To verify: A2 + A = A(A + I),

Given: $A=\left[\begin{array}{ccc}1& 0& -3\\ 2& 1& 3\\ 0& 1& 1\end{array}\right]$

LHS:

RHS:

Therefore, LHS = RHS.

Hence, A2 + A = A(A + I) is verified.

#### Question 53:

If $A=\left[\begin{array}{cc}3& -5\\ -4& 2\end{array}\right]$, then find A2 − 5A − 14I. Hence, obtain A3.

Given: $A=\left[\begin{array}{cc}3& -5\\ -4& 2\end{array}\right]$

Therefore, A2 − 5A − 14I = 0       ...(1)

Premultiplying the (1) by A, we get

A(A2 − 5A − 14I) = A.0
⇒ A3 − 5A2 − 14= 0
⇒ A3 = 5A2 + 14A

#### Question 54:

(i) If $P\left(x\right)=\left[\begin{array}{cc}\mathrm{cos}x& \mathrm{sin}x\\ -\mathrm{sin}x& \mathrm{cos}x\end{array}\right]$, then show that P(x) P(y) = P(x + y) = P(y) P(x).

(ii) If

(i) Given: $P\left(x\right)=\left[\begin{array}{cc}\mathrm{cos}x& \mathrm{sin}x\\ -\mathrm{sin}x& \mathrm{cos}x\end{array}\right]$

then, $P\left(y\right)=\left[\begin{array}{cc}\mathrm{cos}y& \mathrm{sin}y\\ -\mathrm{sin}y& \mathrm{cos}y\end{array}\right]$

Now,

Also,

Now,

From (1), (2) and (3), we get

P(xP(y) = P(x + y) = P(yP(x)

(ii) Given:

Now,

Also,

From (4) and (5), we get
$PQ=\left[\begin{array}{ccc}xa& 0& 0\\ 0& yb& 0\\ 0& 0& zc\end{array}\right]=QP$

#### Question 55:

If $A=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$, find A2 − 5A + 4I and hence find a matrix X such that A2 − 5A + 4I + X = 0.

Given: $A=\left[\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$

Now,

Now, A2 − 5A + 4I + = 0
⇒ = −(A2 − 5A + 4I)

#### Question 56:

If $A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$, prove that ${A}^{n}=\left[\begin{array}{cc}1& n\\ 0& 1\end{array}\right]$ for all positive integers n.

We shall prove the result by the principle of mathematical induction on n.

Step 1: If n = 1, by definition of integral powers of matrix, we have
${A}^{1}=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]=A\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$
So, the result is true for n = 1.

Step 2: Let the result be true for n = m. Then,
${A}^{m}=\left[\begin{array}{cc}1& m\\ 0& 1\end{array}\right]$                   ...(1)

Now, we shall show that the result is true for $n=m+1$.
Here,
${A}^{m+1}=\left[\begin{array}{cc}1& m+1\\ 0& 1\end{array}\right]$

By definition of integral power of matrix, we have

This shows that when the result is true for n = m, it is also true for n = m + 1.

Hence, by the principle of mathematical induction, the result is valid for any positive integer n.

#### Question 57:

If $A=\left[\begin{array}{cc}a& b\\ 0& 1\end{array}\right]$, prove that for every positive integer n.

We shall prove the result by the principle of mathematical induction on n.

Step 1: If n = 1, by definition of integral power of a matrix, we have
${A}^{1}=\left[\begin{array}{cc}{a}^{1}& b\left({a}^{1}-1\right)/a-1\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}a& b\\ 0& 1\end{array}\right]=A\phantom{\rule{0ex}{0ex}}$

So, the result is true for n = 1.

Step 2: Let the result be true for n = m. Then,
${A}^{m}=\left[\begin{array}{cc}{a}^{m}& b\left({a}^{m}-1\right)/a-1\\ 0& 1\end{array}\right]$                       ...(1)

Now, we shall show that the result is true for $n=m+1$.
Here,
${A}^{m+1}=\left[\begin{array}{cc}{a}^{m+1}& b\left({a}^{m+1}-1\right)/a-1\\ 0& 1\end{array}\right]$

By definition of integral power of matrix, we have

This shows that when the result is true for n = m, it is also true for n = m +1.

Hence, by the principle of mathematical induction, the result is valid for any positive integer n.

#### Question 58:

If , then prove by principle of mathematical induction that

for all n ∈ N.

We shall prove the result by the principle of mathematical induction on n.

Step 1: If n = 1, by definition of integral power of a matrix, we have

Thus, the result is true for n=1.

Step 2: Let the result be true for n = m. Then,

Now we shall show that the result is true for $n=m+1$.
Here,
...(1)

By definition of integral power of matrix, we have

This shows that when the result is true for n = m, it is true for $n=m+1$.
Hence, by the principle of mathematical induction, the result is valid for all n$\in N$.

Disclaimer: n is missing before $\theta$ in a12 in An.

#### Question 59:

If , prove that

for all nN.

We shall prove the result by the principle of mathematical induction on n.

Step 1:  If n = 1, by definition of integral power of a matrix, we have

So, the result is true for n = 1.

Step 2: Let the result be true for n = m. Then,
...(1)

Now we shall show that the result is true for $n=m+1$.
Here,

By definition of integral power of matrix, we have

This show that when the result is true for n = m, it is also true for n = m +1.

Hence, by the principle of mathematical induction, the result is valid for all n$\in N$.

#### Question 60:

Let $A=\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]$. Use the principle of mathematical induction to show that

${A}^{n}=\left[\begin{array}{ccc}1& n& n\left(n+1\right)/2\\ 0& 1& n\\ 0& 0& 1\end{array}\right]$ for every positive integer n.

We shall prove the result by the principle of mathematical induction on n.

Step 1: If n = 1, by definition of integral power of a matrix, we have

${A}^{1}=\left[\begin{array}{ccc}1& 1& 1\left(1+1\right)/2\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]=A\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

Thus, the result is true for n = 1.

Step 2: Let the result be true for n = m. Then,

${A}^{m}=\left[\begin{array}{ccc}1& m& m\left(m+1\right)/2\\ 0& 1& m\\ 0& 0& 1\end{array}\right]$                      ...(1)

Now, we shall show that the result is true for $n=m+1$.
Here,

By definition of integral power of matrix, we have

This shows that when the result is true for n = m, it is also true for n = m + 1.

Hence, by the principle of mathematical induction, the result is valid for any positive integer n.

#### Question 61:

If B, C are n rowed square matrices and if A = B + C, BC = CB, C2 = O, then show that for every nN, An+1 = Bn (B + (n + 1) C).

Let $P\left(n\right)$ be the statement given by .

For n = 1, we have

Hence, the statement is true for n = 1.

If the statement is true for n = k, then
...(1)

For $P\left(k+1\right)$ to be true, we must have

Now,

So the statement is true for n = k+1.
Hence, by the principle of mathematical induction, $P\left(n\right)$ is true for all $n\in N$.

#### Question 62:

If A = diag (a, b, c), show that An = diag (an, bn, cn) for all positive integer n.

We shall prove the result by the principle of mathematical induction on n.

Step 1: If n = 1, by definition of integral power of a matrix, we have

${A}^{1}=\left[\begin{array}{ccc}{a}^{1}& 0& 0\\ 0& {b}^{1}& 0\\ 0& 0& {c}^{1}\end{array}\right]=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]=A\phantom{\rule{0ex}{0ex}}$

So, the result is true for n = 1.

Step 2: Let the result be true for n = m. Then,

${A}^{m}=\left[\begin{array}{ccc}{a}^{m}& 0& 0\\ 0& {b}^{m}& 0\\ 0& 0& {c}^{m}\end{array}\right]$                           ...(1)

Now, we shall check if the result is true for $n=m+1$.
Here,
${A}^{m+1}=\left[\begin{array}{ccc}{a}^{m+1}& 0& 0\\ 0& {b}^{m+1}& 0\\ 0& 0& {c}^{m+1}\end{array}\right]$

By definition of integral power of matrix, we have

This shows that when the result is true for n = m, it is also true for $n=m+1$.
Hence, by the principle of mathematical induction, the result is valid for any positive integer n.

#### Question 63:

If A is a square matrix, using mathematical induction prove that (AT)n = (An)T for all n ∈ ℕ.

Let the given statement P(n), be given as
P(n): (AT)n = (An)T for all n ∈ ℕ.

We observe that
P(1): (AT)1 = AT = (A1)T
Thus, P(n) is true for n = 1.

Assume that P(n) is true for n = k ∈ ℕ.
i.e., P(k): (AT)k = (Ak)T

To prove that P(k + 1) is true, we have
(AT)k + 1 = (AT)k.(AT)1
= (Ak)T.(A1)T
= (A+ 1)T
Thus, P(k + 1) is true, whenever P(k) is true.

Hence, by the Principle of mathematical induction, P(n) is true for all n ∈ ℕ.

#### Question 64:

A matrix X has a + b rows and a + 2 columns while the matrix Y has b + 1 rows and a + 3 columns. Both matrices XY and YX exist. Find a and b. Can you say XY and YX are of the same type? Are they equal.

Since the order of the matrices XY and YX is not same, XY and YX are not of the same type and they are unequal.

#### Question 65:

Give examples of matrices
(i) A and B such that ABBA
(ii) A and B such that AB = O but A ≠ 0, B ≠ 0.
(iii) A and B such that AB = O but BAO.
(iv) A, B and C such that AB = AC but BC, A ≠ 0.

Thus, AB ≠ BA.

Thus, AB = O while A ≠ 0 and B ≠ 0.

Thus, AB = O but BAO.

Thus,
AB = AC
But B ≠ C and A ≠ 0.

#### Question 66:

Let A and B be square matrices of the same order. Does (A + B)2 = A2 + 2AB + B2 hold? If not, why?

We know that a matrix does not have commutative property. So,
ABBA
Thus,
${\left(A+B\right)}^{2}$${A}^{2}+2AB+{B}^{2}$

#### Question 67:

If A and B are square matrices of the same order, explain, why in general
(i) (A + B)2A2 + 2AB + B2
(ii) (A B)2A2 − 2AB + B2
(iii) (A + B) (AB) ≠ A2B2.

We know that a matrix does not have commutative property. So,
ABBA
Thus,
${\left(A+B\right)}^{2}$${A}^{2}+2AB+{B}^{2}$

We know that a matrix does not have commutative property. So,
ABBA
Thus,
${\left(A-B\right)}^{2}$${A}^{2}-2AB+{B}^{2}$

We know that a matrix does not have commutative property. So,
ABBA
Thus,
$\left(A+B\right)\left(A-B\right)$${A}^{2}-{B}^{2}$

#### Question 68:

Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2 B2? Give reasons.

Yes, (AB)2 = A2 B2 if AB = BA.

If AB = BA, then
(AB)2 = (AB)(AB)
= A(BA)B      (associative law)
= A(AB)B
= A2 B2

#### Question 69:

If A and B are square matrices of the same order such that AB = BA, then show that (A + B)2 = A2 + 2AB + B2.

(A + B)2 = (A + B)(A + B)
= A2 + AB + BA B2
= A2 + 2AB + B2          (∵ AB = BA)

Hence, (A + B)2 = A2 + 2AB + B2.

#### Question 70:

Let
Verify that AB = AC though BC, AO.

So, AB = AC though B ≠ C , A ≠ O.

#### Question 71:

Three shopkeepers A, B and C go to a store to buy stationary. A purchases 12 dozen notebooks, 5 dozen pens and 6 dozen pencils. B purchases 10 dozen notebooks, 6 dozen pens and 7 dozen pencils. C purchases 11 dozen notebooks, 13 dozen pens and 8 dozen pencils. A notebook costs 40 paise, a pen costs Rs. 1.25 and a pencil costs 35 paise. Use matrix multiplication to calculate each individual's bill.

 Shopkeepers Notebooks In dozen Pens In dozen Pencils In dozen A 12 5 6 B 10 6 7 C 11 3 8

Here,
Cost of notebooks per dozen = = Rs 4.80
Cost of pens per dozen =  = Rs 15
Cost ofpPencils per dozen =  = Rs 4.20

Thus, the bills of A, B and C are Rs 157.80, Rs 167.40 and Rs 281.40, respectively.

#### Question 72:

The cooperative stores of a particular school has 10 dozen physics books, 8 dozen chemistry books and 5 dozen mathematics books. Their selling prices are Rs. 8.30, Rs. 3.45 and Rs. 4.50 each respectively. Find the total amount the store will receive from selling all the items.

Stock of various types of books in the store is given by

Selling price of various types of books in the store is given by

Total amount received by the store from selling all the items is given by

Required amount = Rs 1597.20

#### Question 73:

In a legislative assembly election, a political group hired a public relations firm to promote its candidates in three ways: telephone, house calls and letters. The cost per contact (in paise) is given matrix A as

The number of contacts of each type made in two cities X and Y is given in matrix B as

Find the total amount spent by the group in the two cities X and Y.

The cost per contact is given by

$A=\left[\begin{array}{c}40\\ 100\\ 50\end{array}\right]\begin{array}{c}\mathrm{Telephone}\\ \mathrm{Housecall}\\ \mathrm{Letter}\end{array}$

The number of contacts of each type made in the two cities X and Y is given by

Total amount spent by the group in the two cities X and Y is given by

Thus,
Amount spent on X = Rs 3400
Amount spent on Y = Rs 7200

#### Question 74:

A trust fund has Rs 30000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30000 among the two types of bonds. If the trust fund must obtain an annual total interest of
(i) Rs 1800 (ii) Rs 2000

If Rs x are invested in the first type of bond and Rs $\left(30000-x\right)$ are invested in the second type of bond, then the matrix $A=\left[\begin{array}{cc}x& 30000-x\end{array}\right]$ represents investment and the matrix $B=\left[\begin{array}{c}\frac{5}{100}\\ \frac{7}{100}\end{array}\right]$ represents rate of interest.

Thus,
Amount invested in the first bond = Rs 15000

Amount invested in the second bond = Rs $\left(30000-15000\right)$
= Rs 15000

Thus,
Amount invested in the first bond = Rs 5000

Amount invested in the second bond = Rs $\left(30000-5000\right)$
= Rs 25000

#### Question 75:

To promote making of toilets for women, an organisation tried to generate awarness through (i) house calls, (ii) letters, and (iii) announcements. The cost for each mode per attempt is given below:
(i) ₹50       (ii) ₹20       (iii) ₹40

The number of attempts made in three villages XY and Z are given below:
(i)               (ii)              (iii)
X      400              300             100
Y      300              250               75
Z      500              400             150

Find the total cost incurred by the organisation for three villages separately, using matrices.

According to the question,

Let A be the matrix showing number of attempts made in three villages XY and Z.
$A=\left[\begin{array}{ccc}400& 300& 100\\ 300& 250& 75\\ 500& 400& 150\end{array}\right]$

And, B be a matrix showing the cost for each mode per attempt.
$B=\left[\begin{array}{c}50\\ 20\\ 40\end{array}\right]$

Now, the total cost per village will be shown by AB.

Hence, the total cost incurred by the organisation for three villages separately is
X: ₹30,000
Y: ₹23,000
Z: ₹39,000

#### Question 76:

There are 2 families A and B. There are 4 men, 6 women and 2 children in family A, and 2 men, 2 women and 4 children in family B. The recommend daily amount of calories is 2400 for men, 1900 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using matrix. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the two families. What awareness can you create among people about the planned diet from this question?

According to the question,

Let X be the matrix showing number of family members in family A and B.

And, Y be a matrix showing the recommend daily amount of calories.
$Y=\left[\begin{array}{c}2400\\ 1900\\ 1800\end{array}\right]$

And, Z be a matrix showing the recommend daily amount of proteins.
$Z=\left[\begin{array}{c}45\\ 55\\ 33\end{array}\right]$

Now, the total requirement of calories of the two families will be shown by XY.

Also, the total requirement of proteins of the two families will be shown by XZ.

Hence, the total requirement of calories and proteins for each of the two families is shown as:

#### Question 77:

In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paisa) is given in matrix A as

The number of contacts of each type made in two cities X and Y is given in the matrix B as

Find the total amount spent by the party in the two cities.

What should one consider before casting his/her vote − party's promotional activity of their social activities?

According to the question,

Let A be the matrix showing the cost per contact (in paisa).

And, B be a matrix showing the number of contacts of each type made in two cities X and Y.

Now, the total amount spent by the party in the two cities will be shown by BA.

Hence, the total amount spent by the party in the two cities is
X: ₹9900
Y: ₹21200

One should consider social activities of a party before casting his/her vote.

#### Question 78:

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves ₹ 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

Let the monthly incomes of Aryan and Babban be 3x and 4x, respectively.

Suppose their monthly expenditures are 5y and 7y, respectively.

Since each saves Rs 15,000 per month,

The above system of equations can be written in the matrix form as follows:

$\left[\begin{array}{cc}3& -5\\ 4& -7\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}15000\\ 15000\end{array}\right]$

or,
AX = B, where

Now,

$\left|\mathrm{A}\right|=\left|\begin{array}{cc}3& -5\\ 4& -7\end{array}\right|=-21-\left(-20\right)=-1$

Adj A=${\left[\begin{array}{cc}-7& -4\\ 5& 3\end{array}\right]}^{T}=\left[\begin{array}{cc}-7& 5\\ -4& 3\end{array}\right]$

So, ${A}^{-1}=\frac{1}{\left|A\right|}adjA=-1\left[\begin{array}{cc}-7& 5\\ -4& 3\end{array}\right]=\left[\begin{array}{cc}7& -5\\ 4& -3\end{array}\right]$

Therefore,

Monthly income of Aryan =

Monthly income of Babban =

From this problem, we are encouraged to understand the power of savings. We should save certain part of our monthly income for the future.

#### Question 79:

A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received ₹ 2800 as interest. However, if trust had interchanged money in bonds, they would have got ₹ 100 less as interes. Using matrix method, find the amount invested by the trust.

Let Rs x be invested in the first bond and Rs y be invested in the second bond.
Let A be the investment matrix and B be the interest per rupee matrix. Then,

If the rates of interest had been interchanged, then the total interest earned is Rs 100 less than the previous interest.

The system of equations (1) and (2) can be expressed as
PX = Q, where
$\left|P\right|=\left|\begin{array}{cc}10& 12\\ 12& 10\end{array}\right|=100-144=-44\ne 0$
Thus, P is invertible.

Therefore, Rs 10,000 be invested in the first bond and Rs 15,000 be invested in the second bond.

#### Question 1:

Let , verify that
(i) (2A)T = 2AT
(ii) (A + B)T = AT + BT
(iii) (AB)T = AT BT
(iv) (AB)T = BT AT

#### Question 2:

If $A=\left[\begin{array}{c}3\\ 5\\ 2\end{array}\right]$ and B = [1 0 4], verify that (AB)T = BT AT

#### Question 3:

Let Find AT, BT and verify that
(i) (A + B)T = AT + BT
(ii) (AB)T = BT AT
(iii) (2A)T = 2AT.

#### Question 4:

If , B = [1 3 −6], verify that (AB)T = BT AT

#### Question 5:

If , find (AB)T

$\mathrm{Here},\phantom{\rule{0ex}{0ex}}AB=\left[\begin{array}{ccc}2& 4& -1\\ -1& 0& 2\end{array}\right]\left[\begin{array}{cc}3& 4\\ -1& 2\\ 2& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒AB=\left[\begin{array}{cc}6-4-2& 8+8-1\\ -3-0+4& -4+0+2\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒AB=\left[\begin{array}{cc}0& 15\\ 1& -2\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒{\left(AB\right)}^{T}=\left[\begin{array}{cc}0& 1\\ 15& -2\end{array}\right]$

#### Question 6:

(i) For two matrices A and B, verify that
(AB)T = BT AT.

(ii) For the matrices A and B, verify that (AB)T = BT AT, where

#### Question 7:

If , find AT − BT.

Given:

${B}^{T}=\left[\begin{array}{cc}-1& 1\\ 2& 2\\ 1& 3\end{array}\right]$

Now,

Therefore, ${A}^{T}-{B}^{T}=\left[\begin{array}{cc}4& 3\\ -3& 0\\ -1& -2\end{array}\right]$.

#### Question 8:

If , then verify that AT A = I2.

Hence proved.

#### Question 9:

If , verify that AT A = I2.

#### Question 10:

If li, mi, nii = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AAT = I, where $A=\left[\begin{array}{ccc}{l}_{1}& {m}_{1}& {n}_{1}\\ {l}_{2}& {m}_{2}& {n}_{2}\\ {l}_{3}& {m}_{3}& {n}_{3}\end{array}\right]$.

Given,
are the direction cosines of three mutually perpendicular vectors in space.

Let $A=\left[\begin{array}{ccc}{l}_{1}& {m}_{1}& {n}_{1}\\ {l}_{2}& {m}_{2}& {n}_{2}\\ {l}_{3}& {m}_{3}& {n}_{3}\end{array}\right]$
$⇒{A}^{T}=\left[\begin{array}{ccc}{l}_{1}& {l}_{2}& {l}_{3}\\ {m}_{1}& {m}_{2}& {m}_{3}\\ {n}_{1}& {n}_{2}& {n}_{3}\end{array}\right]$
$A{A}^{T}=\left[\begin{array}{ccc}{l}_{1}& {m}_{1}& {n}_{1}\\ {l}_{2}& {m}_{2}& {n}_{2}\\ {l}_{3}& {m}_{3}& {n}_{3}\end{array}\right]\left[\begin{array}{ccc}{l}_{1}& {l}_{2}& {l}_{3}\\ {m}_{1}& {m}_{2}& {m}_{3}\\ {n}_{1}& {n}_{2}& {n}_{3}\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒A{A}^{T}=\left[\begin{array}{ccc}{{l}_{1}}^{2}+{{m}_{1}}^{2}+{{n}_{1}}^{2}& {l}_{1}{l}_{2}+{m}_{1}{m}_{2}+{n}_{1}{n}_{2}& {l}_{3}{l}_{1}+{m}_{3}{m}_{1}+{n}_{3}{n}_{1}\\ {l}_{1}{l}_{2}+{m}_{1}{m}_{2}+{n}_{1}{n}_{2}& {{l}_{2}}^{2}+{{m}_{2}}^{2}+{{n}_{2}}^{2}& {l}_{2}{l}_{3}+{m}_{2}{m}_{3}+{n}_{2}{n}_{3}\\ {l}_{3}{l}_{1}+{m}_{3}{m}_{1}+{n}_{3}{n}_{1}& {l}_{2}{l}_{3}+{m}_{2}{m}_{3}+{n}_{2}{n}_{3}& {{l}_{3}}^{2}+{{m}_{3}}^{2}+{{n}_{3}}^{2}\end{array}\right]\phantom{\rule{0ex}{0ex}}$
From (i) and (ii), we get
$A{A}^{T}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=I$
Hence proved.

#### Question 1:

If $A=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$, prove that AAT is a skew-symmetric matrix.

#### Question 1:

If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?

We know that if a matrix is of order $m×n$, then it has mn elements.

The possible orders of a matrix with 8 elements are given below:
1$×$8, 2$×$4, 4$×$2, 8$×$

Thus, there are 4 possible orders of the matrix.

The possible orders of a matrix with 5 elements are given below:
1$×$5, 5$×$1

Thus, there are 2 possible orders of the matrix.

#### Question 2:

If A = [aij] = and B = [bij] =
then find (i) a22 + b21 (ii) a11 b11 + a22 b22

$\left(i\right)$

${a}_{22}+{b}_{21}$

$\left(ii\right)\phantom{\rule{0ex}{0ex}}$

${a}_{11}{b}_{11}+{a}_{22}{b}_{22}$

#### Question 3:

Let A be a matrix of order 3 × 4. If R1 denotes the first row of A and C2 denotes its second column, then determine the orders of matrices R1 and C2

The order of ${R}_{1}$ is $1×4$ and the order of .

#### Question 2:

If $A=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]$, show that AAT is a skewsymmetric matrix.

#### Question 3:

If the matrix is a symmetric matrix, find x, y, z and t.

#### Question 4:

Let Find matrices X and Y such that X + Y = A, where X is a symmetric and Y is a skew-symmetric matrix.

#### Question 5:

Express the matrix as the sum of a symmetric and a skew-symmetric matrix.

#### Question 6:

Define a symmetric matrix. Prove that for $A=\left[\begin{array}{cc}2& 4\\ 5& 6\end{array}\right]$, A + AT is a symmetric matrix where AT is the transpose of A.

#### Question 7:

Express the matrix $A=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix.

#### Question 8:

Express the following matrix as the sum of a symmetric and skew-symmetric matrix and verify your result: .

#### Question 9:

For the matrix $A=\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right]$, find A + AT and verify that it is a symmetric matrix.

The given matrix is

$A=\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right]$               .....(1)

$\therefore {A}^{T}=\left[\begin{array}{cc}2& 5\\ 3& 7\end{array}\right]$        ......(2)

Adding (1) and (2), we get

$A+{A}^{T}=\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right]+\left[\begin{array}{cc}2& 5\\ 3& 7\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒A+{A}^{T}=\left[\begin{array}{cc}2+2& 3+5\\ 5+3& 7+7\end{array}\right]=\left[\begin{array}{cc}4& 8\\ 8& 14\end{array}\right]$

A matrix X is said to be symmetric matrix if ${X}^{T}=X$.

Now,

${\left(A+{A}^{T}\right)}^{T}={\left[\begin{array}{cc}4& 8\\ 8& 14\end{array}\right]}^{T}=\left[\begin{array}{cc}4& 8\\ 8& 14\end{array}\right]\phantom{\rule{0ex}{0ex}}\therefore {\left(A+{A}^{T}\right)}^{T}=A+{A}^{T}$

Thus, the matrix $A+{A}^{T}$ is symmetric matrix.

#### Question 1:

If , then A2 is equal to
(a) a null matrix
(b) a unit matrix
(c) −A
(d) A

$\left(b\right)$ a unit matrix

${A}^{2}=AA\phantom{\rule{0ex}{0ex}}⇒{A}^{2}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒{A}^{2}=\left[\begin{array}{ccc}1+0+0& 0+0+0& 0+0-0\\ 0+0+0& 0+1+0& 0+0-0\\ a+0-a& 0+b-b& 0+0+1\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒{A}^{2}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

#### Question 2:

If $A=\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]$, nN, then A4n equals

(a) $\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$

(b) $\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

(c) $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

(d) $\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$

(c) $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

So, $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ is repeated on multiple of 4 and 4n is a multiple of 4.

Thus,
${A}^{4n}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

#### Question 3:

If A and B are two matrices such that AB = A and BA = B, then B2 is equal to
(a) B
(b) A
(c) 1
(d) 0

(a) B

#### Question 4:

If AB = A and BA = B, where A and B are square matrices, then
(a) B2 = B and A2 = A
(b) B2B and A2 = A
(c) A2A, B2 = B
(d) A2A, B2B

(a) B2 = B and A2 = A

#### Question 5:

If A and B are two matrices such that AB = B and BA = A, A2 + B2 is equal to
(a) 2 AB
(b) 2 BA
(c) A + B
(d) AB

(c) A + B

#### Question 6:

If , then the least positive integral value of k is
(a) 3
(b) 4
(c) 6
(d) 7

(d) 7

Now we check if the pattern is same for k = 6.
Here,

Now, we check if the pattern is same for k = 7.
Here,

So, the least positive integral value of k is 7.

#### Question 7:

If the matrix AB is zero, then
(a) It is not necessary that either A = O or, B = O
(b) A = O or B = O
(c) A = O and B = O
(d) all the above statements are wrong

(a) It is not necessary that either A = O or, B = O

#### Question 8:

Let A = $\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]$, then An is equal to

(a) $\left[\begin{array}{ccc}{a}^{n}& 0& 0\\ 0& {a}^{n}& 0\\ 0& 0& a\end{array}\right]$

(b) $\left[\begin{array}{ccc}{a}^{n}& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]$

(c) $\left[\begin{array}{ccc}{a}^{n}& 0& 0\\ 0& {a}^{n}& 0\\ 0& 0& {a}^{n}\end{array}\right]$

(d) $\left[\begin{array}{ccc}na& 0& 0\\ 0& na& 0\\ 0& 0& na\end{array}\right]$

(c) $\left[\begin{array}{ccc}{a}^{n}& 0& 0\\ 0& {a}^{n}& 0\\ 0& 0& {a}^{n}\end{array}\right]$

#### Question 9:

If A, B are square matrices of order 3, A is non-singular and AB = O, then B is a
(a) null matrix
(b) singular matrix
(c) unit-matrix
(d) non-singular matrix

$\left(a\right)$ null matrix

Since A is non-singular matrix and the determinant of a non-singular matrix is non-zero, B should be a null matrix.

#### Question 10:

If , then AB is equal to
(a) B
(b) nB
(c) Bn
(d) A + B

(b) nB

#### Question 11:

If $A=\left[\begin{array}{cc}1& a\\ 0& 1\end{array}\right]$, then An (where nN) equals

(a) $\left[\begin{array}{cc}1& na\\ 0& 1\end{array}\right]$

(b) $\left[\begin{array}{cc}1& {n}^{2}a\\ 0& 1\end{array}\right]$

(c) $\left[\begin{array}{cc}1& na\\ 0& 0\end{array}\right]$

(d) $\left[\begin{array}{cc}n& na\\ 0& n\end{array}\right]$

(a) $\left[\begin{array}{cc}1& na\\ 0& 1\end{array}\right]$

This pattern is applicable for all natural numbers.

#### Question 12:

If and AB = I3, then x + y equals
(a) 0
(b) −1
(c) 2
(d) none of these

(a) 0

#### Question 13:

If and (A + B)2 = A2 + B2, values of a and b are
(a) a = 4, b = 1
(b) a = 1, b = 4
(c) a = 0, b = 4
(d) a = 2, b = 4

(b) a = 1, b = 4

#### Question 14:

If is such that A2 = I, then
(a) 1 + α2 + βγ = 0
(b) 1 − α2 + βγ = 0
(c) 1 − α2 − βγ = 0
(d) 1 + α2 − βγ = 0

(c) 1 − α2 − βγ = 0

#### Question 15:

If S = [Sij] is a scalar matrix such that sij = k and A is a square matrix of the same order, then AS = SA = ?
(a) Ak
(b) k + A
(c) kA
(d) kS

(c) kA

Here,

#### Question 16:

If A is a square matrix such that A2 = A, then (I + A)3 − 7A is equal to
(a) A
(b) IA
(c) I
(d) 3A

(c) I

#### Question 17:

If a matrix A is both symmetric and skew-symmetric, then
(a) A is a diagonal matrix
(b) A is a zero matrix
(c) A is a scalar matrix
(d) A is a square matrix

(b) A is a zero matrix

Let $A=\left[{a}_{ij}\right]$ be a matrix which is both symmetric and skew-symmetric.

If $A=\left[{a}_{ij}\right]$ is a symmetric matrix, then
${a}_{ij}={a}_{ji}$ for all i, j                   ...(1)

If $A=\left[{a}_{ij}\right]$ is a  skew-symmetric matrix, then
${a}_{ij}=-{a}_{ji}$ for all i, j
$⇒{a}_{ji}=-{a}_{ij}$ for all i,j            ...(2)

From eqs. (1) and (2), we have

#### Question 18:

The matrix is
(a) a skew-symmetric matrix
(b) a symmetric matrix
(c) a diagonal matrix
(d) an uppertriangular matrix

(a) a skew-symmetric matrix

Here,

A = $\left[\begin{array}{ccc}0& 5& -7\\ -5& 0& 11\\ 7& -11& 0\end{array}\right]$

$⇒$AT = $\left[\begin{array}{ccc}0& -5& 7\\ 5& 0& -11\\ -7& 11& 0\end{array}\right]$

$⇒{A}^{T}=-\left[\begin{array}{ccc}0& 5& -7\\ -5& 0& 11\\ 7& -11& 0\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒{A}^{T}=-A$

Thus, A is a skew-symmetric matrix.

#### Question 19:

If A is a square matrix, then AA is a
(a) skew-symmetric matrix
(b) symmetric matrix
(c) diagonal matrix
(d) none of these

(d) none of these

Given: A is a square matrix.

#### Question 20:

If A and B are symmetric matrices, then ABA is
(a) symmetric matrix
(b) skew-symmetric matrix
(c) diagonal matrix
(d) scalar matrix

(a) symmetric matrix

#### Question 21:

If $A=\left[\begin{array}{cc}5& x\\ y& 0\end{array}\right]$ and A = AT, then
(a) x = 0, y = 5
(b) x + y = 5
(c) x = y
(d) none of these

(c) x = y

#### Question 22:

If A is 3 × 4 matrix and B is a matrix such that A'B and BA' are both defined. Then, B is of the type
(a) 3 × 4
(b) 3 × 3
(c) 4 × 4
(d) 4 × 3

(a) 3 × 4

The order of A is 3 $×$ 4. So, the order of A' is 4 $×$ 3.

Now, both are defined. So, the number of columns in A' should be equal to the number of rows in B for A'B.
Also, the number of columns in B should be equal to number of rows in A' for BA'.

Hence, the order of matrix B is 3 $×$ 4.

#### Question 23:

If A = [aij] is a square matrix of even order such that aij = i2j2, then
(a) A is a skew-symmetric matrix and | A | = 0
(b) A is symmetric matrix and | A | is a square
(c) A is symmetric matrix and | A | = 0
(d) none of these.

(d) none of these

#### Question 24:

If , then AT + A = I2, if
(a) θ = n π, n ∈ Z
(b) θ = (2n + 1)$\frac{\mathrm{\pi }}{2}$, nZ
(c) θ = 2n π + $\frac{\mathrm{\pi }}{3}$, n ∈ Z
(d) none of these

(c) θ = 2nπ + $\frac{\mathrm{\pi }}{3}$, n ∈ Z

#### Question 25:

If is expressed as the sum of a symmetric and skew-symmetric matrix, then the symmetric matrix is

(a)

(b)

(c)

(d) $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

(a)

#### Question 26:

Out of the given matrices, choose that matrix which is a scalar matrix:

(a) $\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

(b) $\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]$

(c) $\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\right]$

(d) $\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

A diagonal matrix in which all the diagonal elements are equal is called the scalar matrix.

#### Question 27:

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is
(a) 27
(b) 18
(c) 81
(d) 512

(d) 512

There are 9 elements in a 3$×$3 matrix and one element can be filled in two ways, either with 0 or 1.

Thus,
Total possible matrices = ${2}^{9}$ = 512

#### Question 28:

Which of the given values of x and y make the following pairs of matrices equal?

(a) x = $-\frac{1}{3}$, y = 7
(b) y = 7, x = $-\frac{2}{3}$
(c) x = $-\frac{1}{3}$, 4 = $-\frac{2}{5}$
(d) Not possible to find

(d) Not possible to find

#### Question 29:

If and , then the values of k, a, b, are respectively
(a) −6, −12, −18
(b) −6, 4, 9
(c) −6, −4, −9
(d) −6, 12, 18

(c) −6, −4, −9

#### Question 30:

If , then B equals
(a) I cos θ + J sin θ
(b) I sin θ + J cos θ
(c) I cos θ − J sin θ
(d) −I cos θ + J sin θ