Question

If A, B are two matrix then prove that
$(AB{)}^{-1}=B^{-1}{A}^{-1}$

Answer 1

Let $A,B$ be two $3\times 3$ matrices.

To prove: $(AB{)}^{-1}=B^{-1}{A}^{-1}$

Let $I$ be a $3\times 3$ identity matrix.

$\left(AB\right)(AB{)}^{-1}=I$

Premultiply with $A^{-1}$ on both sides

$\Rightarrow \left(A^{-1}AB\right)(AB{)}^{-1}=A^{-1}I$

$\Rightarrow \left(IB\right)(AB{)}^{-1}=A^{-1}$

$\Rightarrow B(AB{)}^{-1}=A^{-1}$

Premultiply with $B^{-1}$ on both sides

$\Rightarrow B^{-1}B(AB{)}^{-1}=B^{-1}{A}^{-1}$

$\Rightarrow I(AB{)}^{-1}=B^{-1}{A}^{-1}$

$\Rightarrow (AB{)}^{-1}=B^{-1}{A}^{-1}$