Question

If $A=\left[\begin{array}{cc}0& -1\\ 4& -3\end{array}\right]$, $B=\left[\begin{array}{c}-5\\ 6\end{array}\right]$ and $3A\times M=2B$
find matrix $M.$

Answer 1

$A=\left[\begin{array}{cc}0& -1\\ 4& -3\end{array}\right]B=\left[\begin{array}{c}-5\\ 6\end{array}\right]3A={\left[\begin{array}{cc}0& -3\\ 12& -9\end{array}\right]}_{2\times 2}2B={\left[\begin{array}{c}-10\\ 12\end{array}\right]}_{2\times 1}$

So dimension of $M$ is $2\times 1$

Let's assume $M=\left[\begin{array}{c}a\\ b\end{array}\right]$

$3A\times M=2B$

$\left[\begin{array}{cc}0& -3\\ 12& -9\end{array}\right]\times \left[\begin{array}{c}a\\ b\end{array}\right]=\left[\begin{array}{c}-10\\ 12\end{array}\right]$

or $\left[\begin{array}{c}-3b\\ 12a-9b\end{array}\right]=\left[\begin{array}{c}-10\\ 12\end{array}\right]$

or $-3b=-10b=\frac{10}{3}$

$12a-9b=12$

or $12a-30=12$

or $12a=42$

or $a=\frac{42}{12}=\frac{7}{2}M=\left[\begin{array}{c}\frac{7}{2}\\ \frac{10}{3}\end{array}\right]$