Theorem 1 If A be any given square matrix of order n, then
A(adj A) = (adj A) A = A I ,
where I is the identity matrix of order n
Verification
Let
A = 
11
12
13
21
22
23
31
32
33
a
a
a
a
a
a
a
a
a










,  then adj  A = 
11
21
31
12
22
32
13
23
33
A
A
A
A
A
A
A
A
A










Since sum of product of elements of a row (or a column) with corresponding
cofactors is equal to |A| and otherwise zero, we have
Rationalised 2023-24
DETERMINANTS     89
A (adj A) = 
A
0
0
0
A
0
0
0
A










 = A  
1
0
0
0
1
0
0
0
1










 = A  I
Similarly, we can show  (adj A) A = A  I
Hence A (adj A) = (adj A) A = A  I
Definition 4 A square matrix A is said to be singular if A  = 0 For example, the determinant of matrix A = 
1
2
4
8



  is zero
Hence A is a singular matrix Definition 5 A square matrix A is said to be non-singular if A  ≠ 0
Let
A = 
1
2
3
4





 Then A = 
1
2
3
4  = 4 – 6 = – 2 ≠ 0