RD Chapter 7 Adjoint and Inverse of a Matrix Ex 7.1 Solutions
Question - 1 : - Find the adjoint of each of the following matrices:Verifythat (adj A) A = |A| I = A (adj A) for the above matrices.
Answer - 1 : -
(i) Let
A =
Cofactors of A are
C11 = 4
C12 = – 2
C21 = – 5
C22 = – 3
(ii) Let
A = 
Therefore cofactors of A are
C11 = d
C12 = – c
C21 = – b
C22 = a
(iii) Let
A = 
Therefore cofactors of A are
C11 = cos α
C12 = – sin α
C21 = – sin α
C22 = cos α
(iv) Let
A = 
Therefore cofactors of A are
C11 = 1
C12 = tan α/2
C21 = – tan α/2
C22 = 1
Question - 2 : - Compute the adjoint of each of the following matrices.
Answer - 2 : -
(i) Let
A = 
Therefore cofactors of A are
C11 = – 3
C21 = 2
C31 = 2
C12 = 2
C22 = – 3
C23 = 2
C13 = 2
C23 = 2
C33 = – 3
(ii) Let
A = 
Cofactors of A
C11 = 2
C21 = 3
C31 = – 13
C12 = – 3
C22 = 6
C32 = 9
C13 = 5
C23 = – 3
C33 = – 1
(iii) Let
A = 
Therefore cofactors of A
C11 = – 22
C21 = 11
C31 = – 11
C12 = 4
C22 = – 2
C32 = 2
C13 = 16
C23 = – 8
C33 = 8
(iv) Let
A = 
Therefore cofactors of A
C11 = 3
C21 = – 1
C31 = 1
C12 = – 15
C22 = 7
C32 = – 5
C13 = 4
C23 = – 2
C33 = 2
Answer - 3 : -
Given
A = 
Therefore cofactors of A
C11 = 30
C21 = 12
C31 = – 3
C12 = – 20
C22 = – 8
C32 = 2
C13 = – 50
C23 = – 20
C33 = 5
Question - 4 : - 
Answer - 4 : -
Given
A = 
Cofactors of A
C11 = – 4
C21 = – 3
C31 = – 3
C12 = 1
C22 = 0
C32 = 1
C13 = 4
C23 = 4
C33 = 3
Question - 5 : - 
Answer - 5 : -
Given
A = 
Cofactors of A are
C11 = – 3
C21 = 6
C31 = 6
C12 = – 6
C22 = 3
C32 = – 6
C13 = – 6
C23 = – 6
C33 = 3
Question - 6 : - 
Answer - 6 : -
Given
A = 
Cofactors of A are
C11 = 9
C21 = 19
C31 = – 4
C12 = 4
C22 = 14
C32 = 1
C13 = 8
C23 = 3
C33 = 2
Question - 7 : - Find theinverse of each of the following matrices:
Answer - 7 : -
(i) The criteria of existence of inverse matrix is thedeterminant of a given matrix should not equal to zero.
Now, |A| = cos θ (cos θ) +sin θ (sin θ)
= 1
Hence, A – 1 exists.
Cofactors of A are
C11 = cos θ
C12 = sin θ
C21 = – sin θ
C22 = cos θ
(ii) The criteria of existence of inverse matrix isthe determinant of a given matrix should not equal to zero.
Now, |A| = – 1 ≠ 0
Hence, A – 1 exists.
Cofactors of A are
C11 = 0
C12 = – 1
C21 = – 1
C22 = 0
(iii) The criteria of existence of inverse matrix isthe determinant of a given matrix should not equal to zero.
(iv) The criteria of existence of inverse matrix isthe determinant of a given matrix should not equal to zero.
Now, |A| = 2 + 15 = 17 ≠ 0
Hence, A – 1 exists.
Cofactors of A are
C11 = 1
C12 = 3
C21 = – 5
C22 = 2
Question - 8 : - Find the inverse of each of the following matrices.
Answer - 8 : -
(i) The criteria of existence of inverse matrix is thedeterminant of a given matrix should not equal to zero.
|A| = 
= 1(6 – 1) – 2(4 – 3) + 3(2 – 9)
= 5 – 2 – 21
= – 18≠ 0
Hence, A – 1 exists
Cofactors of A are
C11 = 5
C21 = – 1
C31 = – 7
C12 = – 1
C22 = – 7
C32 = 5
C13 = – 7
C23 = 5
C33 = – 1
(ii) The criteria of existence of inverse matrix isthe determinant of a given matrix should not equal to zero.
|A| = 
= 1 (1 + 3) – 2 (– 1 + 2) + 5 (3 + 2)
= 4 – 2 + 25
= 27≠ 0
Hence, A – 1 exists
Cofactors of A are
C11 = 4
C21 = 17
C31 = 3
C12 = – 1
C22 = – 11
C32 = 6
C13 = 5
C23 = 1
C33 = – 3
(iii) The criteria of existence of inverse matrix isthe determinant of a given matrix should not equal to zero.
|A| = 
= 2(4 – 1) + 1(– 2 + 1) + 1(1 – 2)
= 6 – 2
= – 4≠ 0
Hence, A – 1 exists
Cofactors of A are
C11 = 3
C21 = 1
C31 = – 1
C12 = + 1
C22 = 3
C32 = 1
C13 = – 1
C23 = 1
C33 = 3
(iv) The criteria of existence of inverse matrix isthe determinant of a given matrix should not equal to zero.
|A| = 
= 2(3 – 0) – 0 – 1(5)
= 6 – 5
= 1≠ 0
Hence, A – 1 exists
Cofactors of A are
C11 = 3
C21 = – 1
C31 = 1
C12 = – 15
C22 = 6
C32 = – 5
C13 = 5
C23 = – 2
C33 = 2
(v) The criteria of existence of inverse matrix is thedeterminant of a given matrix should not equal to zero.
|A| = 
= 0 – 1 (16 – 12) – 1 (– 12 + 9)
= – 4 + 3
= – 1≠ 0
Hence, A – 1 exists
Cofactors of A are
C11 = 0
C21 = – 1
C31 = 1
C12 = – 4
C22 = 3
C32 = – 4
C13 = – 3
C23 = 3
C33 = – 4
(vi) The criteria of existence of inverse matrix isthe determinant of a given matrix should not equal to zero.
|A| = 
= 0 – 0 – 1(– 12 + 8)
= 4≠ 0
Hence, A – 1 exists
Cofactors of A are
C11 = – 8
C21 = 4
C31 = 4
C12 = 11
C22 = – 2
C32 = – 3
C13 = – 4
C23 = 0
C33 = 0
(vii) The criteria of existence of inverse matrix isthe determinant of a given matrix should not equal to zero.
|A| = 
– 0 + 0
= – (cos2 α – sin2 α)
= – 1≠ 0
Hence, A – 1 exists
Cofactors of A are
C11 = – 1
C21 = 0
C31 = 0
C12 = 0
C22 = – cos α
C32 = – sin α
C13 = 0
C23 = – sin α
C33 = cos α
Question - 9 : - Find the inverse of each of thefollowing matrices and verify that A-1A = I3.
Answer - 9 : -
(i) We have
|A| = 
= 1(16 – 9) – 3(4 – 3) + 3(3 – 4)
= 7 – 3 – 3
= 1≠ 0
Hence, A – 1 exists
Cofactors of A are
C11 = 7
C21 = – 3
C31 = – 3
C12 = – 1
C22 = 1
C32 = 0
C13 = – 1
C23 = 0
C33 = 1
(ii) We have
|A| = 
= 2(8 – 7) – 3(6 – 3) + 1(21 – 12)
= 2 – 9 + 9
= 2≠ 0
Hence, A – 1 exists
Cofactors of A are
C11 = 1
C21 = 1
C31 = – 1
C12 = – 3
C22 = 1
C32 = 1
C13 = 9
C23 = – 5
C33 = – 1
For thefollowing pair of matrices verify that (AB)-1 = B-1A-1.
Answer - 10 : -
(i) Given
Hence, (AB)-1 = B-1A-1
(ii) Given
Hence, (AB)-1 = B-1A-1