Mathematics - Matrix Question with Solution | TestHub

We know that, $M(\mathrm{adj}M)=|M|$ I Replacing $\mathrm{M}$ by adj $\mathrm{M},$ we get $\mathrm{adj}M[\mathrm{adj}(\mathrm{adj}M)=\det (\mathrm{adj}M)I$ $=\det (M)M^{-1}[\mathrm{adj}(\mathrm{adj}M)={\alpha }^{2}1$ $$\left[\because {\mathrm{M}}^{-1}=\frac{1}{|\mathrm{M}|}\mathrm{adj}(\mathrm{M})\right]$$ $\Rightarrow \alpha {\mathrm{M}}^{-1}[\mathrm{adj}(\mathrm{adj}M)]={\alpha }^2\mathrm{I}$ $\Rightarrow {\mathrm{M}}^{-1}[\mathrm{adj}(\mathrm{adj}\mathrm{M})]=\alpha \mathrm{I}$ But $M^{-1}[\mathrm{adj}(\mathrm{adj}M)]=KI$ Hence, $\mathrm{K}=\alpha$