Question

If $A=\begin{bmatrix} 2 & 1 & -1 \\ 3 & 5 & 2 \\ 1 & 6 & 1 \end{bmatrix}$ , then $tr (A adj (adj A)$ is equal to (where, $tr(P)$ denotes the trace of the matrix $P$ i.e. the sum of all the diagonal elements of the matrix $P$ and $adj(P)$ denotes the adjoint of matrix $P$ )

• A. $7$
• B. $18$
• C. $-58$
• D. $-1624$

Answer 1

Answer: (D)

$\therefore adj\left(a d j A\right)=\left|A\right|A$ (by property)
$Aadj\left(a d j A\right)=\left|A\right|A^{2}\ldots ..\left(i\right)$
$\left|A\right|=\begin{vmatrix} 2 & 1 & -1 \\ 3 & 5 & 2 \\ 1 & 6 & 1 \end{vmatrix}=2\left(5 - 12\right)-1\left(3 - 2\right)-1\left(18 - 5\right)$
$=-14-1-13=-28$
$A^{2}=\begin{bmatrix} 2 & 1 & -1 \\ 3 & 5 & 2 \\ 1 & 6 & 1 \end{bmatrix}\begin{bmatrix} 2 & 1 & -1 \\ 3 & 5 & 2 \\ 1 & 6 & 1 \end{bmatrix}=\begin{bmatrix} 6 & 1 & -1 \\ 23 & 40 & 9 \\ 21 & 37 & 12 \end{bmatrix}$
$\operatorname{tr}(A(a d j(a d j A)))=\operatorname{tr}\left(|A| A^{2}\right)=-28[6+40+12]$
$=-28\times 58=-1624$