For a 33 matrix M, let trace(M) denote the sum of its diagonal elements.

Question

For a $3	imes3$ matrix $M$, let $\operatorname{trace}(M)$ denote the sum of its diagonal elements. Let $A$ be a $3	imes3$ matrix such that $|A|=	frac12$ and $\operatorname{trace}(A)=3$. If $B=\operatorname{adj}(\operatorname{adj}(2A))$, then the value of $|B|+\operatorname{trace}(B)$ equals:

Accepted Answer

$\det(2A)=2^3\det(A)=8\cdot	frac12=4$.

For a $3	imes3$ matrix, $\operatorname{adj}(\operatorname{adj}(2A))=\det(2A)\cdot(2A)=4\cdot(2A)=8A$. So $B=8A$.

$\det(B)=\det(8A)=8^3\det(A)=512\cdot	frac12=256$.

$\operatorname{trace}(B)=8\operatorname{trace}(A)=8	imes3=24$.

$|B|+\operatorname{trace}(B)=256+24=280$.

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