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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 is/are $ALWAYS TRUE$ ?
$(A)$ $M=I$ $(B)$ $\operatorname{det} M =1$ $(C)$ $M ^2= I$ $(D)$ $(\operatorname{adj} M)^2=I$
$B,C,D$
$A,B,D$
$A,B$
$A,C$
Solution
$\operatorname{det}( M ) \neq 0$
$M ^{-1}=\operatorname{adj}(\operatorname{adj} M )$
$M ^{-1}=\operatorname{det}( M ) \cdot M$ $. . . . . . .(I)$
$M ^{-1} M =\operatorname{det}( M ) \cdot M ^2$
$I =\operatorname{det}( M ) \cdot M ^2$ $. . . . . . .(II)$
$\operatorname{det}( I )=(\operatorname{det}( M ))^5$
$1=\operatorname{det}( M )$
From (i) $I = M ^2$
$(\operatorname{adj} M )^2=\operatorname{adj}\left( M ^2\right)=\operatorname{adj} I = I$
