Mathematics - Matrix Question with Solution | TestHub

Given ${\text{M}}^{-1}=\text{adj}\left(\text{adjM}\right)$

$\text{∴ adj M}\cdot {\text{M}}^{-1}=\text{adj M }\text{. adj }\left(\text{adjM}\right)$

$\text{⇒adj M}.{\text{M}}^{-1}=\left|\text{AdjM}\right|\text{I}$

$\text{⇒adj M}={\left|\text{M}\right|}^2\text{M} .........\left(1\right)$

$\Rightarrow \left|adjM\right|=\left|{\left|M\right|}^{2}M\right|={\left|M\right|}^6\left|M\right|$

$\Rightarrow {\left|M\right|}^2={\left|M\right|}^7\Rightarrow \left|M\right|\ne 0, \left|M\right|=1 .............\left(2\right)$

By equations $\left(1\right)\& \left(2\right)$, we get

$\text{adj M}=\text{M} ...........\left(3\right)$

$\text{⇒M}.\text{adjM}={\text{M}}^2$

$\Rightarrow \left|M\right|I=M^2\Rightarrow M^2=I$

Now, by equation $\left(3\right)$,

$\text{adj M}=\text{M}$

$\Rightarrow {\left(\text{adj }M\right)}^2=M^2=I$