Question

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

• A. $M=I$
• B. $detM=1$
• C. $M^2=I$
• D. ${(adjM)}^2=I$

Answer 1

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

${(adjM)}^2=I$

Explanation for the correct option:

Finding which of the given options are true.

Given:

$M$ is a $3\times 3$ invertible matrix with real entries and $I$is a $3\times 3$ identity matrix

Using Matrix identities, we get

$\begin{array}{rcl}{M}^{-1}& =& adj.(adj.M)={\left|M\right|}^{3-2}.M\\ M^{-1}& =& \left|M\right|M\\ M.M^{-1}& =& I=\left|M\right|M^2\\ & \Rightarrow & \left|I\right|={\left|M\right|}^5[\because Mis3\times 3\text{matrix}]\\ & \Rightarrow & \left|M\right|=\left|I\right|=1...\left(B\right)\end{array}$

So, option (B) is correct now,

$\begin{array}{rcl}& \Rightarrow & Det{M}^{-1}=M\\ & \Rightarrow & M^2=I...\left(C\right)\end{array}$

So, option (C) is also correct now,

$\begin{array}{rcl}{M}^{-1}& =& \frac{adj.M}{\left|M\right|}\Rightarrow adj.M=M\\ {(adj.M)}^2& =& I...\left(D\right)\end{array}$

So, option (D) is also correct

Therefore, the correct answer is options (B) ,(C) and (D)