$\text{ Let }\mathrm{M}\text{ be a }3\times 3\text{ invertible matrix with real entries and let I denote the }3\times 3\text{ identity matrix. If }$

${\mathrm{M}}^{-1}=\text{ adj (adj M), then which of the following statement is/are ALWAYS TRUE ? }$

$\begin{array}{l}\det \left(\mathrm{M}\right)\ne 0\\ {\mathrm{M}}^{-1}=\mathrm{adj}(\mathrm{adj}\mathrm{M})\\ {\mathrm{M}}^{-1}=\det \left(\mathrm{M}\right)\cdot \mathrm{M} \left(\mathrm{since} \mathrm{adj}\left(\mathrm{adj} \mathrm{A}\right)={\left|\mathrm{A}\right|}^{\mathrm{n}-2}\mathrm{A}\right)\\ {\mathrm{M}}^{-1}\mathrm{M}=\det \left(\mathrm{M}\right)\cdot {\mathrm{M}}^2\\ \mathrm{I}=\det \left(\mathrm{M}\right)\cdot {\mathrm{M}}^2\dots \left(i\right)\\ \det \left(\mathrm{I}\right)=(\det (\mathrm{M}){)}^5 \left(\sin ce \left|kA\right|=k^3\left|A\right| if A is 3\times 3 matrix\right)\\ 1=\det \left(\mathrm{M}\right) \cdots \left(ii\right)\\ \text{ From (i) }\mathrm{I}={\mathrm{M}}^2\\ (\mathrm{adj}\mathrm{M}{)}^2=\mathrm{adj}\left({\mathrm{M}}^2\right)=\mathrm{adj}\mathrm{I}=\mathrm{I}\\ \end{array}$