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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 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 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 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 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 1:

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

#### 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 1:

If A is an m × n matrix and B is n × p matrix does AB exist? If yes, write its order.

Given: Order of A = $m×n$
Order of B = $n×p$

Since  the number of columns in A are equal to the number of rows in B, i.e. n, AB exists.
Order of AB = Number of rows in A$×$ Number of columns in B
= $m×p$

#### Question 2:

If . Write the orders of AB and BA.

The order of matrix A is $2×3$ and the order of matrix B is $3×2$.

Since the number of columns in A is equal to the number of rows in B, AB exists and it is of order $2×2$.
Also, since the number of columns in B is equal to the number of rows in A, BA exists and it is of order $3×3$.

#### Question 3:

If , write AB.

$AB=\left[\begin{array}{cc}4& 3\\ 1& 2\end{array}\right]\left[\begin{array}{c}-4\\ 3\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒AB=\left[\begin{array}{c}-16+9\\ -4+6\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒AB=\left[\begin{array}{c}-7\\ 2\end{array}\right]$

#### Question 4:

If $A=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$, write AAT.

#### Question 5:

Given an example of two non-zero 2 × 2 matrices A and B such that AB = O.

#### Question 6:

If $A=\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right]$, find A + AT.

#### Question 7:

If $A=\left[\begin{array}{cc}i& 0\\ 0& i\end{array}\right]$, write A2.

#### Question 8:

If , find x satisfying 0 < x < $\frac{\mathrm{\pi }}{2}$ when A + AT = I

If , find AAT

#### Question 10:

If = I, where I is 2 × 2 unit matrix. Find x and y.

#### Question 11:

If , satisfies the matrix equation A2 = kA, write the value of k.

#### Question 12:

If $A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ satisfies A4 = λA, then write the value of λ.

#### Question 13:

If , find A2.

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

If , find A3.

#### Question 15:

If , find A4.

$\mathrm{Here},\phantom{\rule{0ex}{0ex}}{A}^{2}=AA\phantom{\rule{0ex}{0ex}}⇒{A}^{2}=\left[\begin{array}{cc}-3& 0\\ 0& -3\end{array}\right]\left[\begin{array}{cc}-3& 0\\ 0& -3\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒{A}^{2}=\left[\begin{array}{cc}9+0& 0+0\\ 0+0& 0+9\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒{A}^{2}=\left[\begin{array}{cc}9& 0\\ 0& 9\end{array}\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Now},\phantom{\rule{0ex}{0ex}}{A}^{4}={A}^{2}{A}^{2}\phantom{\rule{0ex}{0ex}}⇒{A}^{4}=\left[\begin{array}{cc}9& 0\\ 0& 9\end{array}\right]\left[\begin{array}{cc}9& 0\\ 0& 9\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒{A}^{4}=\left[\begin{array}{cc}81+0& 0+0\\ 0+0& 0+81\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒{A}^{4}=\left[\begin{array}{cc}81& 0\\ 0& 81\end{array}\right]$

#### Question 16:

If [x  2] $\left[\begin{array}{c}3\\ 4\end{array}\right]=2$, find x

$\mathrm{Given}: \left[\begin{array}{cc}x& 2\end{array}\right]\left[\begin{array}{c}3\\ 4\end{array}\right]=2\phantom{\rule{0ex}{0ex}}⇒\begin{array}{c}3x+8\end{array}=2\phantom{\rule{0ex}{0ex}}⇒3x=2-8\phantom{\rule{0ex}{0ex}}⇒3x=-6\phantom{\rule{0ex}{0ex}}⇒x=\frac{-6}{3}\phantom{\rule{0ex}{0ex}}⇒x=-2$

#### Question 17:

If A = [aij] is a 2 × 2 matrix such that aij = i + 2j, write A.

#### Question 18:

Write matrix A satisfying

#### Question 19:

If A = [aij] is a square matrix such that aij = i2j2, then write whether A is symmetric or skew-symmetric.

#### Question 20:

For any square matrix write whether AAT is symmetric or skew-symmetric.

Here,

Thus, AAT is a symmetric matrix.

#### Question 21:

If A = [aij] is a skew-symmetric matrix, then write the value of $\sum _{i}$ aij.

#### Question 22:

If A = [aij] is a skew-symmetric matrix, then write the value of $\sum _{i}\sum _{j}$ aij.

#### Question 23:

If A and B are symmetric matrices, then write the condition for which AB is also symmetric.

Given: AB is symmetric.

Thus, AB is also symmetric, if AB = BA.

#### Question 24:

If B is a skew-symmetric matrix, write whether the matrix AB AT is symmetric or skew-symmetric.

If B is a skew-symmetric matrix, then ${B}^{T}=-B$.

#### Question 25:

If B is a symmetric matrix, write whether the matrix AB AT is symmetric or skew-symmetric.

If B is a symmetric matrix, then ${B}^{T}=B$.

$\therefore$ is a symmetric matrix.

#### Question 26:

If A is a skew-symmetric and nN such that (An)T = λAn, write the value of λ.

Given: A is skew symmetric matrix.

$⇒{A}^{T}=-A$

#### Question 27:

If A is a symmetric matrix and nN, write whether An is symmetric or skew-symmetric or neither of these two.

Hence, ${A}^{n}$ is a symmetric matrix.

#### Question 28:

If A is a skew-symmetric matrix and n is an even natural number, write whether An is symmetric or skew-symmetric or neither of these two.

Hence, ${A}^{n}$ is symmetric when n is an even natural number.

#### Question 29:

If A is a skew-symmetric matrix and n is an odd natural number, write whether An is symmetric or skew-symmetric or neither of the two.

Hence, ${A}^{n}$ is skew-symmetric when n is an odd natural number.

#### Question 30:

If A and B are symmetric matrices of the same order, write whether ABBA is symmetric or skew-symmetric or neither of the two.

Since A and B are symmetric matrices, .

Here,

#### Question 31:

Write a square matrix which is both symmetric as well as skew-symmetric.

#### Question 32:

Find the values of x and y, if $2\left[\begin{array}{cc}1& 3\\ 0& x\end{array}\right]+\left[\begin{array}{cc}y& 0\\ 1& 2\end{array}\right]=\left[\begin{array}{cc}5& 6\\ 1& 8\end{array}\right]$

#### Question 33:

If , find x and y

The corresponding elements of two equal matrices are equal.

#### Question 34:

Find the value of x from the following:

The corresponding elements of two equal matrices are equal.

#### Question 35:

Find the value of y, if

#### Question 36:

Find the value of x, if

The corresponding elements of two equal matrices are equal.

#### Question 37:

If matrix A = [1 2 3], write AAT.

#### Question 38:

If , then find x.

The corresponding elements of two equal matrices are equal.

#### Question 39:

If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, find A + AT.

#### Question 40:

If , then find a.

The corresponding elements of two equal matrices are equal.

#### Question 41:

If A is a matrix of order 3 × 4 and B is a matrix of order 4 × 3, find the order of the matrix of AB.

If A is a matrix of order 3 × 4 and B is a matrix of order 4 × 3, then the order of matrix AB is given by the number of rows in A and number of columns in B, respectively.

Thus, the order of matrix AB is $3×3$.

#### Question 42:

If is identity matrix, then write the value of α.

#### Question 43:

If $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{cc}3& 1\\ 2& 5\end{array}\right]=\left[\begin{array}{cc}7& 11\\ k& 23\end{array}\right]$, then write the value of k.

#### Question 44:

If I is the identity matrix and A is a square matrix such that A2 = A, then what is the value of (I + A)2 = 3A?

Given: A is a square matrix, such that ${A}^{2}=A$.

Here,

#### Question 45:

If $A=\left[\begin{array}{cc}1& 2\\ 0& 3\end{array}\right]$ is written as B + C, where B is a symmetric matrix and C is a skew-symmetric matrix, then B is equal to.