Page 98 - DMTH201_Basic Mathematics-1

Unit 4: Determinants

Notes
Example: For the following matrices find A 1 and verify that (i) A (Adj A) = (Adj A) A
= |A| I and (ii) AA 1 A 1 A I

1 1 1
1 1
1. 2. 2 1 0
2 2
3 2 1
Solution:

1 1
1. Let A
2 2

1 1
A
| | 2 2 4 0
2 2
Cofactor of 1 = + (2) = 2
Cofactor of 1 = (2) = 2 I column
Cofactor of 2 = ( 1) = 1
Cofactor of 2 = + (1) = 1 II column

2 1
Adj A
2 1

1 1
Adj A 1 2 1 2 4
A 1
A
| | 4 2 1 1 1
2 4

1 1 2 1 2 2 1 1 4 0
A (Adj A )
2 2 2 1 4 4 2 2 0 4

1 0

dj
I
A (A A) 4 4I A ( A 4)
0 1
I
Similarly it can be verified that (Adj ) A .
A
A
1 1 1 2 1 1 2 2 1 1
Now, AA 1
2 2 4 2 1 4 4 4 2 2
1 4 0 1 0
. I
4 0 4 0 1

AA 1 . I
1
Similarly, it can be verified that A A . I

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