Question

State whether the following statement is True or False:
${\left(AB\right)}^{-1}=A^{-1}{B}^{-1}$,where $A$ and $B$ are invertible matrices satisfying commutative property with respect to multiplication.

• A. True
• B. False

Answer 1

The correct option is A True

We have
${\left(AB\right)}^{-1}=A^{-1}{B}^{-1}$
Pre multiply by $AB$ on both sides
$\Rightarrow AB{\left(AB\right)}^{-1}=AB\left(A^{-1}{B}^{-1}\right)$
$[\because A{A}^{-1}=I]$
$\Rightarrow I=AB\left(A^{-1}{B}^{-1}\right)$ ...(1)
Taking $\text{RHS}=AB\left(A^{-1}{B}^{-1}\right)$
$=\left(BA\right)\left(A^{-1}{B}^{-1}\right)$
$[\because AB=BA]$
$=B\left(A{A}^{-1}\right)B^{-1}$
$\text{RHS}=BI{B}^{-1}$ $[\because A{A}^{-1}=I]$
$\text{RHS}=B{B}^{-1}$
$\text{RHS}=I$
From equation (1) and (2)
$\text{LHS}=\text{RHS}$
$\Rightarrow (AB{)}^{-1}=A^{-1}{B}^{-1}$, when $AB=BA$
Hence, the given statement is true.