(2)
Thus
B = BI = B (AC) = (BA) C = IC = C
Theorem 4 If A and B are invertible matrices of the same order, then (AB)–1 = B–1 A–1 Proof From the definition of inverse of a matrix, we have
(AB) (AB)–1 = 1
or
A–1 (AB) (AB)–1 = A–1I
(Pre multiplying both sides by A–1)
or
(A–1A) B (AB)–1 = A–1
(Since A–1 I = A–1)
or
IB (AB)–1 = A–1
or
B (AB)–1 = A–1
or
B–1 B (AB)–1 = B–1 A–1
or
I (AB)–1 = B–1 A–1
Hence
(AB)–1 = B–1 A–1
1 Matrices A and B will be inverse of each other only if
     (A) AB = BA (B) AB = BA = 0
     (C) AB = 0, BA = I (D) AB = BA = I
EXERCISE 3 4
Rationalised 2023-24
 70
MATHEMATICS
Miscellaneous Examples
Example 23 If 
cos
sin
A
sin
cos
θ
θ


= 

−
θ
θ


, then prove that 
cos
sin
A
sin
cos
n
n
n
n
n
θ
θ


= 

−
θ
θ


, n ∈ N