Question

If A=$\left|\begin{array}{ccc}3& -1& 0\\ -1& 2& 3\end{array}\right|$ and B=$\left|\begin{array}{ccc}-1& 1& 2\\ 0& 2& -1\end{array}\right|$ then find ${\left(A{B}^T\right)}^T$

• A. $\left|\begin{array}{cc}2& -5\\ 1& 6\end{array}\right|$
• B. $\left|\begin{array}{cc}3& -4\\ 7& -6\end{array}\right|$
• C. $\left|\begin{array}{cc}-4& 9\\ -2& 1\end{array}\right|$
• D. $\left|\begin{array}{cc}-3& 8\\ 4& 2\end{array}\right|$

Answer 1

The correct option is C $\left|\begin{array}{cc}-4& 9\\ -2& 1\end{array}\right|$

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

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

$=\left|\begin{array}{cc}-4& -2\\ 9& 1\end{array}\right|$
And $(A{B}^T{)}^T=\left|\begin{array}{cc}-4& 9\\ -2& 1\end{array}\right|$