Question

Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M^{-1}=\text{adj}(\text{adj} M)$, then which of the following statements is/are ALWAYS TRUE?

• A. $M=I$
• B. $\text{det} M=1$
• C. $M^{2}=I$
• D. $(\text{adj} M)^{2}=I$

Answer 1

Answer: (B), (C), (D)

$\because M^{-1}=\text{adj}(\text{adj} M)$
$\Rightarrow M^{-1}=|M| M \ldots(1)$
$\Rightarrow |M-1|=\left|M^{3}\right||M|$
$\Rightarrow |M|^{5}=1 \Rightarrow |M|=1$
Substituting $|M|=1$ in $(1)$ we get
$M=M^{-1} \Rightarrow M^{2}=1$
also adjM $=|M| \cdot M^{-1}=M$
Hence $(\text{adj} M)^{2}=M^{2}=I$