# If A=((3,2), (-3,-4)) and B=((0,-5),(-2,1)), What are the matrices X and Y such that 2A -3X = B and 3A+2Y = 2B?

Jan 18, 2018

$X = \left(\begin{matrix}2 & 3 \\ - \frac{4}{3} & - 3\end{matrix}\right)$ and $Y = \left(\begin{matrix}- \frac{9}{2} & - 8 \\ \frac{5}{2} & 7\end{matrix}\right)$

#### Explanation:

As $2 A - 3 X = B$, $3 X = 2 A - B$ and

$X = \frac{1}{3} \left(2 A - B\right)$

= $\frac{1}{3} \left[2 \left(\begin{matrix}3 & 2 \\ - 3 & - 4\end{matrix}\right) - \left(\begin{matrix}0 & - 5 \\ - 2 & 1\end{matrix}\right)\right]$

= $\frac{1}{3} \left(\begin{matrix}2 \times 3 - 0 & 2 \times 2 - \left(- 5\right) \\ 2 \times \left(- 3\right) - \left(- 2\right) & 2 \times \left(- 4\right) - 1\end{matrix}\right)$

= $\frac{1}{3} \left(\begin{matrix}6 & 9 \\ - 4 & - 9\end{matrix}\right)$

= $\left(\begin{matrix}2 & 3 \\ - \frac{4}{3} & - 3\end{matrix}\right)$

and as $3 A + 2 Y = 2 B$, $2 Y = 2 B - 3 A$

and $Y = \frac{1}{2} \left(2 B - 3 A\right)$

= $\frac{1}{2} \left[2 \left(\begin{matrix}0 & - 5 \\ - 2 & 1\end{matrix}\right) - 3 \left(\begin{matrix}3 & 2 \\ - 3 & - 4\end{matrix}\right)\right]$

= $\frac{1}{2} \left(\begin{matrix}2 \times 0 - 3 \times 3 & 2 \times \left(- 5\right) - 3 \times 2 \\ 2 \times \left(- 2\right) - 3 \times \left(- 3\right) & 2 \times 1 - 3 \times \left(- 4\right)\end{matrix}\right)$

= $\frac{1}{2} \left(\begin{matrix}- 9 & - 16 \\ 5 & 14\end{matrix}\right)$

= $\left(\begin{matrix}- \frac{9}{2} & - 8 \\ \frac{5}{2} & 7\end{matrix}\right)$