Question

If $A=\left[\begin{array}{cc}2& 0\\ 5& -3\end{array}\right]$ and $B=\left[\begin{array}{cc}-2& 1\\ 3& -1\end{array}\right]$, then find the trace of $(A{B}^T{)}^T$.

Answer 1

$A=\left[\begin{array}{cc}2& 0\\ 5& -3\end{array}\right]$ and $B=\left[\begin{array}{cc}-2& 1\\ 3& -1\end{array}\right]$

$B^T=\left[\begin{array}{cc}-2& 3\\ 1& -1\end{array}\right]$

$A\times B^T=\left[\begin{array}{cc}2& 0\\ 5& -3\end{array}\right]\left[\begin{array}{cc}-2& 3\\ 1& -1\end{array}\right]$

$=\left[\begin{array}{cc}-4+0& 6-0\\ -10-3& 15+3\end{array}\right]=\left[\begin{array}{cc}-4& 6\\ -13& 18\end{array}\right]$

$(A{B}^T{)}^T=\left[\begin{array}{cc}-4& -13\\ 6& 18\end{array}\right]$

Trace of $(A{B}^T{)}^T=-4+18=14$