Find a matrix A such that …

According to the question we  are required to find matrix A such that {tex} \left[ \begin{array} { c c } { 2 } & { - 1 } \\ { 1 } & { 0 } \\ { - 3 } & { 4 } \end{array} \right] A = \left[ \begin{array} { c c } { - 1 } & { - 8 } \\ { 1 } & { - 2 } \\ { 9 } & { 22 } \end{array} \right]{/tex} .

Let us take the   order of  matrix A as m{tex}\times{/tex}n {tex}{/tex},therefore, m = 2, n = 2.
Let {tex}A = \left[ \begin{array} { l l } { x } & { y } \\ { s } & { t } \end{array} \right]{/tex} ...(i)
{tex}\therefore \left[ \begin{array} { c c } { 2 } & { - 1 } \\ { 1 } & { 0 } \\ { - 3 } & { 4 } \end{array} \right] A = \left[ \begin{array} { c c } { - 1 } & { - 8 } \\ { 1 } & { - 2 } \\ { 9 } & { 22 } \end{array} \right]{/tex}
{tex}\Rightarrow \left[ \begin{array} { c c } { 2 } & { - 1 } \\ { 1 } & { 0 } \\ { - 3 } & { 4 } \end{array} \right] \left[ \begin{array} { l l } { x } & { y } \\ { s } & { t } \end{array} \right] = \left[ \begin{array} { c c } { - 1 } & { - 8 } \\ { 1 } & { - 2 } \\ { 9 } & { 22 } \end{array} \right]{/tex}
{tex}\Rightarrow \left[ \begin{array} { c c } { 2 x - s } & { 2 y - t } \\ { x } & { y } \\ { - 3 x + 4 s } & { - 3 y + 4 t } \end{array} \right] = \left[ \begin{array} { c c } { - 1 } & { - 8 } \\ { 1 } & { - 2 } \\ { 9 } & { 22 } \end{array} \right]{/tex}
Therefore,on equating corresponding elements both sides, we get,
2 - s = -1, x = 1, y = -2 and 2y - t = -8
At x = 1, 2x - s = -1 {tex}\Rightarrow{/tex} 2{tex}\times{/tex}1 - s = -1
{tex}\Rightarrow{/tex} -s = -1 - 2 {tex}\Rightarrow{/tex} s = 3 and at y = -2, 2y - t = -8,
{tex}\Rightarrow{/tex} 2{tex}\times{/tex}(-2) - t = -8 {tex}\Rightarrow{/tex} -4 - t = -8 {tex}\Rightarrow{/tex} t = 4
Therefore,on putting x = 1, y = -2, s = 3 and t = 4 in Eq. (i),
we get, {tex}A = \left[ \begin{array} { c c } { 1 } & { - 2 } \\ { 3 } & { 4 } \end{array} \right]{/tex}