Since B is the inverse of A
AB = BA = I (1)
Since C is also the inverse of A
AC = CA = I (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
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