Question

Find matrix $A$ such that $\left[\begin{array}{cc}2& -1\\ 1& 0\\ -3& 4\end{array}\right]A=\left[\begin{array}{cc}-1& -8\\ 1& -2\\ 9& 22\end{array}\right]$

Answer 1

Given : $\left[\begin{array}{cc}2& -1\\ 1& 0\\ -3& 4\end{array}\right]A=\left[\begin{array}{cc}-1& -8\\ 1& -2\\ 9& 22\end{array}\right]$

$A$ should have $2$ rows, since it is multiplied with $3\times 2$ and $2$ columns as it gives the result as $3\times 2$ matrix.

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$⟹\left[\begin{array}{cc}2& -1\\ 1& 0\\ -3& 4\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}-1& -8\\ 1& -2\\ 9& 22\end{array}\right]$

$⟹\left[\begin{array}{cc}2a-c& 2b-d\\ a+0& b+0\\ -3a+4c& -3b+4d\end{array}\right]=\left[\begin{array}{cc}-1& -8\\ 1& -2\\ 9& 22\end{array}\right]$

$⟹2a-c=-1$ ........ $\left(i\right)$

$2b-d=-8$ .......... $\left(ii\right)$

$a+0=1,b+0=-2$

$⟹a=1$ and $b=-2$

Substitute $a$ in $\left(i\right)$ we get

$2\left(1\right)-c=-1⟹c=3$

Substitute $b$ in $\left(ii\right)$ we get

$2(-2)-d=-8⟹d=4$

Now, substitute the values of $a,b,c$ and $d$ in $A$ we get

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