Question

If for any matrix $M,M^{-1}$ exists, then which of the following is not true?

• A. $\left|M^{-1}\right|=|M{|}^{-1}$
• B. ${\left(M^2\right)}^{-1}\ne {\left(M^{-1}\right)}^2$
• C. ${\left(M^{\prime }\right)}^{-1}={\left(M^{-1}\right)}^{\prime }$
• D. ${\left(M^{-1}\right)}^{-1}=M$

Answer 1

Answer: (B)

All are true but ${\left(M^2\right)}^{-1}\ne {\left(M^{-1}\right)}^2$ is not true.
Since, ${\left(M^2\right)}^{-1}=(M\cdot M{)}^{-1}\left(\because M^2=M\cdot M\right)$
$=M^{-1}\cdot M^{-1}\left[(AB{)}^{-1}=B^{-1}{A}^{-1}\right]$
$={\left(M^{-1}\right)}^2$
$\therefore {\left(M^2\right)}^{-1}={\left(M^{-1}\right)}^2$