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7. Adjoint and Inverse of a Matrix
Exercise 7.1
1 A. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = 4
C
12
 = – 2
C
21
 = – 5
C
22
 = – 3
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
(adj A)A = 
And, |A|.I = 
Also, A(adj A) = 
A(adj A) = 
Hence, (adj A)A = |A|.I = A.(adj A)
1 B. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Page 2


7. Adjoint and Inverse of a Matrix
Exercise 7.1
1 A. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = 4
C
12
 = – 2
C
21
 = – 5
C
22
 = – 3
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
(adj A)A = 
And, |A|.I = 
Also, A(adj A) = 
A(adj A) = 
Hence, (adj A)A = |A|.I = A.(adj A)
1 B. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = d
C
12
 = – c
C
21
 = – b
C
22
 = a
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
(adj A)A = 
And, |A|.I = 
Also, A(adj A) = 
Hence, (adj A)A = |A|.I = A.(adj A)
1 C. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = 
C
12
 = 
C
21
 = 
C
22
 = 
Since, adj A = 
(adj A) = 
= 
Page 3


7. Adjoint and Inverse of a Matrix
Exercise 7.1
1 A. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = 4
C
12
 = – 2
C
21
 = – 5
C
22
 = – 3
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
(adj A)A = 
And, |A|.I = 
Also, A(adj A) = 
A(adj A) = 
Hence, (adj A)A = |A|.I = A.(adj A)
1 B. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = d
C
12
 = – c
C
21
 = – b
C
22
 = a
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
(adj A)A = 
And, |A|.I = 
Also, A(adj A) = 
Hence, (adj A)A = |A|.I = A.(adj A)
1 C. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = 
C
12
 = 
C
21
 = 
C
22
 = 
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
(adj A)A = 
And, |A|.I = 
= 
= 
= 
Also, A(adj A) = 
= 
Hence, (adj A)A = |A|.I = A.(adj A)
1 D. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = 1
C
12
 = 
C
21
 = 
C
22
 = 1
Since, adj A = 
(adj A) = 
= 
Page 4


7. Adjoint and Inverse of a Matrix
Exercise 7.1
1 A. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = 4
C
12
 = – 2
C
21
 = – 5
C
22
 = – 3
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
(adj A)A = 
And, |A|.I = 
Also, A(adj A) = 
A(adj A) = 
Hence, (adj A)A = |A|.I = A.(adj A)
1 B. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = d
C
12
 = – c
C
21
 = – b
C
22
 = a
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
(adj A)A = 
And, |A|.I = 
Also, A(adj A) = 
Hence, (adj A)A = |A|.I = A.(adj A)
1 C. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = 
C
12
 = 
C
21
 = 
C
22
 = 
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
(adj A)A = 
And, |A|.I = 
= 
= 
= 
Also, A(adj A) = 
= 
Hence, (adj A)A = |A|.I = A.(adj A)
1 D. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = 1
C
12
 = 
C
21
 = 
C
22
 = 1
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
= 
(adj A)A = 
And, |A|.I = 
= 
Also, A(adj A) = 
= 
= 
Hence, (adj A)A = |A|.I = A.(adj A)
2 A. Question
Find the adjoint of each of the following Matrices and Verify that (adj A) A = |A| I = A (adj A)
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are:
C
11
 = – 3 C
21
 = 2 C
31
 = 2
C
12
 = 2 C
22
 = – 3 C
23
 = 2
C
13
 = 2 C
23
 = 2 C
33
 = – 3
adj A = 
= 
Page 5


7. Adjoint and Inverse of a Matrix
Exercise 7.1
1 A. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = 4
C
12
 = – 2
C
21
 = – 5
C
22
 = – 3
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
(adj A)A = 
And, |A|.I = 
Also, A(adj A) = 
A(adj A) = 
Hence, (adj A)A = |A|.I = A.(adj A)
1 B. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = d
C
12
 = – c
C
21
 = – b
C
22
 = a
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
(adj A)A = 
And, |A|.I = 
Also, A(adj A) = 
Hence, (adj A)A = |A|.I = A.(adj A)
1 C. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = 
C
12
 = 
C
21
 = 
C
22
 = 
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
(adj A)A = 
And, |A|.I = 
= 
= 
= 
Also, A(adj A) = 
= 
Hence, (adj A)A = |A|.I = A.(adj A)
1 D. Question
Find the adjoint of each of the following Matrices.
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are
C
11
 = 1
C
12
 = 
C
21
 = 
C
22
 = 1
Since, adj A = 
(adj A) = 
= 
Now, (adj A)A = 
= 
(adj A)A = 
And, |A|.I = 
= 
Also, A(adj A) = 
= 
= 
Hence, (adj A)A = |A|.I = A.(adj A)
2 A. Question
Find the adjoint of each of the following Matrices and Verify that (adj A) A = |A| I = A (adj A)
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A are:
C
11
 = – 3 C
21
 = 2 C
31
 = 2
C
12
 = 2 C
22
 = – 3 C
23
 = 2
C
13
 = 2 C
23
 = 2 C
33
 = – 3
adj A = 
= 
Now, (adj A).A = 
= 
= 
Also, |A|.I =  = ( – 3 + 4 + 4)
= 
Then, A.(adj A) = 
= 
= 
Since, (adj A).A = |A|.I = A(adj A)
2 B. Question
Find the adjoint of each of the following Matrices and Verify that (adj A) A = |A| I = A (adj A)
Verify that (adj A) A=|A| I=A (adj A) for the above matrices.
Answer
A = 
Cofactors of A
C
11
 = 2 C
21
 = 3 C
31
 = – 13
C
12
 = – 3 C
22
 = 6 C
32
 = 9
C
13
 = 5 C
23
 = – 3 C
33
 = – 1
adj A = 
=

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