Question

Find the matrix $B$, such that
$\left[\begin{array}{cc}4& 3\\ 3& 2\end{array}\right]B=\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]$

Answer 1

Let $B=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
Then, $\left[\begin{array}{cc}4& 3\\ 3& 2\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}4a+3c& 4b+3d\\ 3a+2c& 3b+2d\end{array}\right]=\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]$
Thus, $4a+3c=2$ ....(i)
$3a+2c=1$ ....(ii)
$4b+3d=5$ ....(iii)
and $3b+2d=3$ .....(iv)
Solving (i) and (ii), we get
$a=-1,c=2$
Solving (iii) and (iv), we get
$b=-1,d=3$
$B=\left[\begin{array}{cc}-1& -1\\ 2& 3\end{array}\right]$