Question

The matrix A satisfying the equation $\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]A=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]$ is

• A. $\left[\begin{array}{cc}1& 4\\ -1& 0\end{array}\right]$
• B. $\left[\begin{array}{cc}1& -4\\ 1& 0\end{array}\right]$
• C. $\left[\begin{array}{cc}1& -2\\ 0& -1\end{array}\right]$
• D. none of these

Answer 1

The correct option is D none of these

$\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]A=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]$
Let $P=\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right],Q=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]$
$PA=Q$
$\Rightarrow A=P^{-1}Q$
We have $P=\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]$
$|P|=1$
Now $adjP=C^T={\left[\begin{array}{cc}1& 0\\ -3& 1\end{array}\right]}^T$
$\Rightarrow adjP=\left[\begin{array}{cc}1& -3\\ 0& 1\end{array}\right]$
Hence, $P^{-1}=\left[\begin{array}{cc}1& -3\\ 0& 1\end{array}\right]$
So, $A=\left[\begin{array}{cc}1& -3\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]$
$\Rightarrow A=\left[\begin{array}{cc}1& 4\\ 0& -1\end{array}\right]$