Let M be a 3 × 3 invertible matrix with real entries and let I...

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

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

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

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

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

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

By equations \left(1\right) \text{ and } \left(2\right)$, we get</p><p>$\text{adj M} = \text{M} . . . . . . . . . . . \left(3\right)$</p><p>$\text{⇒M} . \text{adjM} = {\text{M}}^{2}$</p><p>$⇒ \left|M\right|I = {M}^{2} ⇒ {M}^{2} = I$</p><p>Now, by equation \left(3\right)$,

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

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