Question

If the matrix A is such that $\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]A=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]$, then what is equal to A?

• A. $\left[\begin{array}{cc}1& 4\\ 0& -1\end{array}\right]$
• B. $\left[\begin{array}{cc}1& 4\\ 0& 1\end{array}\right]$
• C. $\left[\begin{array}{cc}-1& 4\\ 0& -1\end{array}\right]$
• D. $\left[\begin{array}{cc}1& -4\\ 0& -1\end{array}\right]$

Answer 1

Answer: (A)

$\left[\begin{array}{cc}1& 4\\ 0& -1\end{array}\right]$

Given

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

Let $A=\left[\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right]$

$\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]\left[\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right]=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]$

$\left[\begin{array}{cc}\alpha +3\gamma & \beta +4\delta \\ \gamma & \delta \end{array}\right]=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]$

Two matrices are equal if corresponding elements are equal

$\Rightarrow \alpha +3\gamma =1,\gamma =0,$

$\beta +3\delta =1$ and $\delta =-1$

$\Rightarrow \alpha =1,$ $\beta =4$

$\Rightarrow A=\left[\begin{array}{cc}1& 4\\ 0& -1\end{array}\right]$