Question

If $A$ is an $3\times 3$ non–singular matrix such that $AA\text{'}=A\text{'}A$ and $B=A^{-1}A\text{'}$, then $BB\text{'}$ equals:

• A. $I+B$
• B. $I$
• C. $B^{-1}$
• D. $(B^{-1}{)}^{\prime }$

Answer 1

The correct option is B $I$

Consider $B=A^{-1}{A}^{\prime }$
$\Rightarrow B^{\prime }=(A^{-1}{A}^{\prime }{)}^{\prime }=A(A^{-1}{)}^{\prime }$
$\Rightarrow B{B}^{\prime }=\left(A^{-1}{A}^{\prime }\right)\left(A\right(A^{-1}{)}^{\prime })$
$=\left(A^{-1}A\right)\left(A^{\prime }\right(A^{\prime }{)}^{-1})$
$=I$