Then A = 
1
2
3
4  = 4 – 6 = – 2 ≠ 0 Hence A is a nonsingular matrix
We state the following theorems without proof Theorem 2 If A and B are nonsingular matrices of the same order, then AB and BA
are also nonsingular matrices of the same order Theorem 3 The determinant of the product of matrices is equal to product of their
respective determinants, that is, AB  = A  B , where A and B are square matrices of
the same order
Remark We know that (adj A) A = A  I = 
A
A
A
A
0
0
0
0
0
0
0







≠
,
Writing determinants of matrices on both sides, we have
(
adjA)A
 =
A
0
0
0
A
0
0
0
A
Rationalised 2023-24
 90
MATHEMATICS
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