For a 3 3 matrix M, let trace (M) denote the sum of all the diagonal elements of M.

Question

For a $3 	imes 3$ matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 	imes 3$ matrix such that $|A|=\frac{1}{2}$ and trace $(A)=3$. If $B=\operatorname{adj}(\operatorname{adj}(2 A))$, then the ...

Accepted Answer

$B=adj(adj(2A))=det(2A)\cdot (2A)$

Since $A$ is a $3	imes 3$ matrix with

$det(A)=\frac{1}{2},$

the determinant of $2A$ is computed as

$det(2A)={2}^{3}det(A)=8\cdot \frac{1}{2}=4.$

Thus,

$B=4\cdot (2A)=8A.$

Now, compute the determinant and the trace of $B$:

Determinant of $B$:

$det(B)=det(8A)={8}^{3}det(A)=512\cdot \frac{1}{2}=256.$

Trace of $B$:

$trace(B)=trace(8A)=8\cdot trace(A)=8\cdot 3=24.$

Finally, adding these results:

$det(B)+trace(B)=256+24=280.$

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