Let M be a 3 times 3 invertible matrix with r | vedclass

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(A) Given $M^{-1} = \operatorname{adj}(\operatorname{adj} M)$.We know that for a $3 	imes 3$ matrix $M$,$\operatorname{adj}(\operatorname{adj} M) = (

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$B, C, D$

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(A) Given $M^{-1} = \operatorname{adj}(\operatorname{adj} M)$.We know that for a $3 	imes 3$ matrix $M$,$\operatorname{adj}(\operatorname{adj} M) = (\operatorname{det} M)^{3-2} M = (\operatorname{det} M) M$.Thus,$M^{-1} = (\operatorname{det} M) M$.Multiplying both sides by $M$,we get $M^{-1} M = (\operatorname{det} M) M^2$,which implies $I = (\operatorname{det} M) M^2$.Taking the determinant on both sides,$\operatorname{det}(I) = \operatorname{det}((\operatorname{det} M) M^2)$.$1 = (\operatorname{det} M)^3 \cdot (\operatorname{det} M)^2 = (\operatorname{det} M)^5$.Since $M$ has real entries,$\operatorname{det} M = 1$.Substituting $\operatorname{det} M = 1$ into $I = (\operatorname{det} M) M^2$,we get $I = M^2$.Since $M^2 = I$,then $(\operatorname{adj} M)^2 = \operatorname{adj}(M^2) = \operatorname{adj}(I) = I$.Thus,statements $B, C,$ and $D$ are always true.

Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M^{-1} = \operatorname{adj}(\operatorname{adj} M)$,then which of the following statement$(s)$ is/are $ALWAYS \text{ } TRUE$?

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