Question

If $A=\left[\begin{array}{ccc}2& 1& -1\\ 3& 5& 2\\ 1& 6& 1\end{array}\right]$ , then $tr(Aadj(adjA)$ 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)

$\therefore adj\left(adjA\right)=\left|A\right|A$ (by property)
$Aadj\left(adjA\right)=\left|A\right|A^2\dots ..\left(i\right)$
$\left|A\right|=\left|\begin{array}{ccc}2& 1& -1\\ 3& 5& 2\\ 1& 6& 1\end{array}\right|=2\left(5-12\right)-1\left(3-2\right)-1\left(18-5\right)$
$=-14-1-13=-28$
$A^2=\left[\begin{array}{ccc}2& 1& -1\\ 3& 5& 2\\ 1& 6& 1\end{array}\right]\left[\begin{array}{ccc}2& 1& -1\\ 3& 5& 2\\ 1& 6& 1\end{array}\right]=\left[\begin{array}{ccc}6& 1& -1\\ 23& 40& 9\\ 21& 37& 12\end{array}\right]$
$\mathrm{tr}(A(adj(adjA)))=\mathrm{tr}\left(|A|A^2\right)=-28[6+40+12]$
$=-28\times 58=-1624$