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)

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