Page No 7.22:
Question 1:
Find the adjoint of each of the following matrices:
(i)
(ii)
(iii)
(iv)
Verify that (adj A) A = |A| I = A (adj A) for the above matrices.
Answer:
Given below are the squares matrices. Here, we will interchange the diagonal elements and change the signs of
the off-diagonal elements.
s.
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Question 2:
Compute the adjoint of each of the following matrices:
(i)
(ii)
(iii)
(iv)
Verify that (adj A) A = |A| I = A (adj A) for the above matrices.
Answer:
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Question 3:
For the matrix , show that A (adj A) = O.
Answer:
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Question 4:
If , show that adj A = A.
Answer:
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Question 5:
If , show that adj A = 3AT.
Answer:
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Question 6:
Find A (adj A) for the matrix
Answer:
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Question 7:
Find the inverse of each of the following matrices:
(i)
(ii)
(iii)
(iv)
Answer:
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Question 8:
Find the inverse of each of the following matrices.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
Answer:
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Question 9:
Find the inverse of each of the following matrices and verify that .
(i)
(ii)
Answer:
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Question 10:
For the following pairs of matrices verity that
(i)
(ii)
Answer:
(AB)−1=B−1 A−1 (AB)−1=B−1 A−1(AB)−1=B−1 A−1
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Question 11:
Let
Answer:
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Question 12:
Given , compute A−1 and show that
Answer:
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Question 13:
If , then show that
Answer:
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Question 14:
Find the inverse of the matrix and show that
Answer:
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Question 15:
Given . Compute (AB)−1.
Answer:
We have,
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Question 16:
Let
Show that
(i)
(ii)
(iii) .
Answer:
​
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Question 17:
If , verify that . Hence, find A−1.
Answer:
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Question 18:
Show that satisfies the equation . Hence, find A−1.
Answer:
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Question 19:
If , show that . Hence, find A−1.
Answer:
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Question 20:
If , find x and y such that . Hence, evaluate A−1.
Answer:
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Question 21:
If , find the value of so that . Hence, find A−1.
Answer:
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Question 22:
Show that satisfies the equation . Thus, find A−1.
Answer:
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Question 23:
Show that satisfies the equation . Thus, find A−1.
Answer:
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Question 24:
For the matrix . Show that . Hence, find A−1.
Answer:
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Question 25:
Show that the matrix, satisfies the equation, . Hence, find A−1.
Answer:
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Question 26:
If . Verify that and hence find A−1.
Answer:
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Question 27:
If , prove that .
Answer:
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Question 28:
If , show that .
Answer:
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Question 29:
If , show that
Answer:
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Question 30:
Solve the matrix equation , where X is a 2 × 2 matrix.
Answer:
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Question 31:
Find the matrix X satisfying the matrix equation
.
Answer:
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Question 32:
Find the matrix X for which
Answer:
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Question 33:
Find the matrix X satisfying the equation
Answer:
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Question 34:
If , find and prove that
Answer:
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Question 35:
Prove that .
Answer:
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Question 36:
Answer:
We know that (AB)−1 = B−1 A−1.
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Question 37:
If
Answer:
We know that (AT)−1 = (A−1)T.
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Question 38:
Find the adjoint of the matrix and hence show that .
Answer:
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Question 39:
Answer:
Page No 7.34:
Question 1:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 2:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 3:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 4:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 5:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 6:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 7:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 8:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 9:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
Page No 7.34:
Question 10:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 11:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 12:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 13:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 14:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 15:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
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Question 16:
Find the inverse of each of the following matrices by using elementary row transformations:
Answer:
Page No 7.35:
Question 1:
Write the adjoint of the matrix
Answer:
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Question 2:
If A is a square matrix such that A (adj A) 5I, where I denotes the identity matrix of the same order. Then, find the value of |A|.
Answer:
We know
Here,
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Question 3:
If A is a square matrix of order 3 such that |A| = 5, write the value of |adj A|.
Answer:
For any square matrix of order n,
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Question 4:
If A is a square matrix of order 3 such that |adj A| = 64, find |A|.
Answer:
For any square matrix of order n,
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Question 5:
If A is a non-singular square matrix such that |A| = 10, find |A−1|.
Answer:
For any non-singular matrix A,
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Question 6:
If A, B, C are three non-null square matrices of the same order, write the condition on A such that AB = AC ⇒ B = C.
Answer:
Consider .
On multiplying both sides by , we get
Therefore, the required condition is A must be invertible or .
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Question 7:
If A is a non-singular square matrix such that , then find
Answer:
For any invertible matrix A,
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Question 8:
If adj , find adj AB.
Answer:
Given:
For any two non-singular matrices,
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Question 9:
If A is symmetric matrix, write whether AT is symmetric or skew-symmetric.
Answer:
For any symmetric matrix, .
Hence, is also symmetric.
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Question 10:
If A is a square matrix of order 3 such that |A| = 2, then write the value of adj (adj A).
Answer:
For any square matrix A, we have
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Question 11:
If A is a square matrix of order 3 such that |A| = 3, then write the value of adj (adj A).
Answer:
For any square matrix A, we have
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Question 12:
If A is a square matrix of order 3 such that adj (2A) = k adj (A), then write the value of k.
Answer:
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Question 13:
If A is a square matrix, then write the matrix adj (AT) − (adj A)T.
Answer:
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Question 14:
Let A be a 3 × 3 square matrix, such that A (adj A) = 2 I, where I is the identity matrix. Write the value of |adj A|.
Answer:
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Question 15:
If A is a non-singular symmetric matrix, write whether A−1 is symmetric or skew-symmetric.
Answer:
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Question 16:
If , then find the value of k.
Answer:
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Question 17:
If A is an invertible matrix such that |A−1| = 2, find the value of |A|.
Answer:
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Question 18:
If A is a square matrix such that , then write the value of |adj A|.
Answer:
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Question 19:
If be such that then find the value of k.
Answer:
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Question 20:
Let A be a square matrix such that , then write interms of A.
Answer:
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Question 21:
If Cij is the cofactor of the element aij of the matrix , then write the value of a32C32.
Answer:
In the given matrix ,
C32 = (−1)3 + 2 (8 − 30) = 22
Therefore, a32C32 = 5 × 22 = 110.
Hence, the value of a32C32 is 110.
Page No 7.36:
Question 22:
Find the inverse of the matrix
Answer:
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Question 23:
Find the inverse of the matrix .
Answer:
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Question 24:
If , write adj A.
Answer:
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Question 25:
If , find adj (AB).
Answer:
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Question 26:
If , then find |adj A|.
Answer:
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Question 27:
If , write in terms of A.
Answer:
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Question 28:
Write
Answer:
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Question 29:
Use elementary column operation C2 → C2 + 2C1 in the following matrix equation :
Answer:
Applying C2 → C2 + 2C1, we get
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Question 30:
In the following matrix equation use elementary operation R2 → R2 + R1 and the equation thus obtained:
Answer:
By applying elementary operation R2 → R2 + R1, we get
(Every row operation is equivalent to left-multiplication by an elementary matrix.)
Page No 7.37:
Question 1:
If A is an invertible matrix, then which of the following is not true
(a)
(b)
(c)
(d)
Answer:
(a)
We have, , and all are the properties of the inverse of a matrix A
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Question 2:
If A is an invertible matrix of order 3, then which of the following is not true
(a)
(b)
(c) If , where B and C are square matrices of order 3
(d)
Answer:
(c) If , then where B and C are square matrices of order 3.
If A is an invertible matrix, then exists.
Now,
On multiplying both sides by , we get
Therefore, the statement ​given in (c) is not true.
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Question 3:
If
(a) is a skew-symmetric matrix
(b) A−1 + B−1
(c) does not exist
(d) none of these
Answer:
(d) none of these
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Question 4:
If , then adj A is
(a)
(b)
(c)
(d)
Answer:
(b)
Adjoint of a square matrix of order 2 is obtained by interchanging the diagonal elements and changing the signs of off-diagonal elements.
Here,
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Question 5:
If A is a singular matrix, then adj A is
(a) non-singular
(b) singular
(c) symmetric
(d) not defined
Answer:
(b) singular
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Question 6:
If A, B are two n × n non-singular matrices, then
(a) AB is non-singular
(b) AB is singular
(c)
(d) (AB)−1 does not exist
Answer:
(a) AB is non-singular
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Question 7:
If , then the value of |adj A| is
(a) a27
(b) a9
(c) a6
(d) a2
Answer:
(c) a6
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Question 8:
If , then ded (adj (adj A)) is
(a) 144
(b) 143
(c) 142
(d) 14
Answer:
(a) 144
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Question 9:
If B is a non-singular matrix and A is a square matrix, then det (B−1 AB) is equal to
(a) Det (A−1)
(b) Det (B−1)
(c) Det (A)
(d) Det (B)
Answer:
(c) Det (A)
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Question 10:
For any 2 × 2 matrix, if , then |A| is equal to
(a) 20
(c) 100
(d) 10
(d) 0
Answer:
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Question 11:
If A5 = O such that equals
(a) A4
(b) A3
(c) I + A
(d) none of these
Answer:
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Question 12:
If A satisfies the equation then A−1 exists if
(a)
(b)
(c)
(d)
Answer:
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Question 13:
If for the matrix A, A3 = I, then A−1 =
(a) A2
(b) A3
(c) A
(d) none of these
Answer:
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Question 14:
If A and B are square matrices such that B = − A−1 BA, then (A + B)2 =
(a) O
(b) A2 + B2
(c) A2 + 2AB + B2
(d) A + B
Answer:
(b)
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Question 15:
If
(a) 5A
(b) 10A
(c) 16A
(d) 32A
Answer:
(c) 16A
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Question 16:
For non-singular square matrix A, B and C of the same order
(a)
(b)
(c)
(d)
Answer:
Disclaimer: In Quesion, We are to find the inverse of . The inverse is missing in the question.
(d)
We have,
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Question 17:
The matrix is a singular matrix, if the value of b is
(a) − 3
(b) 3
(c) 0
(d) non-existent
Answer:
(d) non-existent
For any singular matrix, the value of the determinant is 0.
Here,
Hence, b is non-existent.
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Question 18:
If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is
(a) dn
(b) dn−1
(c) dn+1
(d) d
Answer:
(b) dn−1
We know,
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Question 19:
If A is a matrix of order 3 and |A| = 8, then |adj A| =
(a) 1
(b) 2
(c) 23
(d) 26
Answer:
(d)
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Question 20:
If , then the inverse of A is
(a) A−2
(b) A + I
(c) I − A
(d) A − I
Answer:
(c) I − A
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Question 21:
If A and B are invertible matrices, which of the following statement is not correct.
(a)
(b)
(c)
(d)
Answer:
(c)
We have, , and all are the properites of inverse of a matrix.
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Question 22:
If A is a square matrix such that A2 = I, then A−1 is equal to
(a) A + I
(b) A
(c) 0
(d) 2A
Answer:
(b) A
On multiplying both sides by , we get
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Question 23:
Let and X be a matrix such that A = BX, then X is equal to
(a)
(b)
(c)
(d) none of these.
Answer:
(a)
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Question 24:
If be such that , then k equals
(a) 19
(b) 1/19
(c) − 19
(d) − 1/19
Answer:
(b) 1/19
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Question 25:
If is orthogonal, then x + y =
(a) 3
(b) 0
(c) − 3
(d) 1
Answer:
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Question 26:
If equals
(a) A
(b) − A
(c) ab A
(d) none of these
Answer:
(d) none of these
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Question 27:
If , then
(a)
(b)
(c)
(d) none of these
Answer:
(b)
Page No 7.39:
Question 28:
If a matrix A is such that 3 equal to
(a)
(b)
(c)
(d) none of these
Answer:
(d) none of these
Page No 7.39:
Question 29:
If A is an invertible matrix, then det (A−1) is equal to
(a)
(b)
(c) 1
(d) none of these
Answer:
(b)
We know that for any invertible matrix A, = .
Page No 7.39:
Question 30:
If
(a) , if n is an even natural number
(b) , if n is an odd natural number
(c)
(d) none of these
Answer:
Disclaimer: In all option, the power of A (i.e. n is missing)
(a) , if n is an even natural number
Generally,
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Question 31:
If x, y, z are non-zero real numbers, then the inverse of the matrix , is
(a)
(b)
(c)
(d)
Answer:
(a)
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