Question

$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]$ then $\left|\right(B^T{A}^T{)}^{-1}|$ is equal to

• A. $10$
• B. $\frac{1}{10}$
• C. $1$
• D. $-1$

Answer 1

Answer: (C)

$1$

$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ $B=\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]$

$B^T{A}^T=\left[\begin{array}{cc}2& 3\\ 1& 4\end{array}\right]\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]=\left[\begin{array}{cc}2+6& 6+12\\ 1+8& 3+16\end{array}\right]=\left[\begin{array}{cc}8& 18\\ 9& 19\end{array}\right]$

$det\left(B^T{A}^T\right)=8\times 19-9\times 18=152-162=-10$

$(B^T{A}^T{)}^{-1}=\frac{1}{det\left(B^T{A}^T\right)}adj(B^T{A}^T)=\frac{1}{-10}\left[\begin{array}{cc}19& -18\\ -9& 8\end{array}\right]$

$det\left(\right(B^T{A}^T{)}^{-1})=\frac{19\times 8-18\times 9}{-10}=1$