Question

$A=\left[\begin{array}{ccc}3& a& -1\\ 2& 5& c\\ b& 8& 2\end{array}\right]$ is symmetric and $B=\left[\begin{array}{ccc}d& 3& a\\ b-a& e& -2b-c\\ -2& 6& -f\end{array}\right]$ is skew-symmetric, then sum of diagonal elements of $AB$

Answer 1

Given $A$ is symmetric
$\Rightarrow A^T=A$
$\Rightarrow \left[\begin{array}{ccc}3& 2& b\\ a& 5& 8\\ -1& c& 2\end{array}\right]=\left[\begin{array}{ccc}3& a& -1\\ 2& 5& c\\ b& 8& 2\end{array}\right]$
$\Rightarrow a=2,b=-1,c=8$
and $B$ is skew-symmetric
$\Rightarrow B^T=-B$
$\Rightarrow \left[\begin{array}{ccc}d& b-a& -2\\ 3& e& 6\\ a& -2b-c& -f\end{array}\right]=\left[\begin{array}{ccc}-d& -3& -a\\ a-b& -e& 2b+c\\ 2& -6& f\end{array}\right]$
$\Rightarrow d=-d,f=-f$ and $e=-e$
$\Rightarrow d=f=e=0$
So,
$A=\left[\begin{array}{ccc}3& 2& -1\\ 2& 5& 8\\ -1& 8& 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}0& 3& 2\\ -3& 0& -6\\ -2& 6& 0\end{array}\right]$
$\Rightarrow AB=\left[\begin{array}{ccc}3& 2& -1\\ 2& 5& 8\\ -1& 8& 2\end{array}\right]\left[\begin{array}{ccc}0& 3& 2\\ -3& 0& -6\\ -2& 6& 0\end{array}\right]=\left[\begin{array}{ccc}-4& 3& -6\\ -31& 54& -26\\ -28& 9& -50\end{array}\right]$