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. $\left(M^2\right)^{-1} \neq\left(M^{-1}\right)^2$
• 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 $\left(M^2\right)^{-1} \neq\left(M^{-1}\right)^2$ 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[(A B)^{-1}=B^{-1} A^{-1}\right]} $
$=\left(M^{-1}\right)^2 $
$\therefore \left(M^2\right)^{-1} =\left(M^{-1}\right)^2 $