Question

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

• A. 10
• B. 12
• C. 14
• D. 16

Answer 1

The correct option is C 14

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

$=\left|\begin{array}{cc}2(-2)+0(-1)& 2\left(3\right)+0(-1)\\ 5(-2)-3\left(1\right)& 5\left(3\right)-3(-1)\end{array}\right|$

$=\left|\begin{array}{cc}-4& 6\\ -13& 18\end{array}\right|$
And ${\left(A{B}^T\right)}^T=\left|\begin{array}{cc}-4& -13\\ 6& 18\end{array}\right|$
Trace of a square matrix is the sum of its diagonal elements.
So, Trace of ${\left(A{B}^T\right)}^T=-4+18=14$