Question

Find matrices $A$ and $B$, where
(i) $2A-B=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$ and $A+3B=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$
(ii)$3A-B=\left[\begin{array}{ccc}-1& 2& 1\\ 1& 0& 5\end{array}\right]$ an $A+5B=\left[\begin{array}{ccc}0& 0& 1\\ -1& 0& 0\end{array}\right]$

Answer 1

$\left(1\right)(2A+6B)-(2A-B)=\left[\begin{array}{cc}0& 2\\ -2& 0\end{array}\right]-\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$

$B=\left[\begin{array}{cc}\frac{-1}{7}& \frac{3}{7}\\ \frac{-2}{7}& \frac{-1}{7}\end{array}\right]$

$2A-\left[\begin{array}{cc}\frac{-1}{7}& \frac{3}{7}\\ \frac{-2}{7}& \frac{-1}{7}\end{array}\right]=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$

$A=\left[\begin{array}{cc}\frac{3}{7}& \frac{-2}{7}\\ \frac{-1}{7}& \frac{3}{7}\end{array}\right]$

$\left(2\right)3(A+5B)=3A+15B=\left[\begin{array}{ccc}0& 0& 3\\ -3& 0& 0\end{array}\right]$

$(3A+15B)-(3A-B)=\left[\begin{array}{ccc}0& 0& 3\\ -3& 0& 0\end{array}\right]-\left[\begin{array}{ccc}-1& 2& 1\\ 1& 0& 5\end{array}\right]$

$16B=\left[\begin{array}{ccc}1& -2& 2\\ -4& 0& -5\end{array}\right]$

$B=\left[\begin{array}{ccc}\frac{1}{16}& \frac{-2}{16}& \frac{2}{16}\\ \frac{-4}{16}& 0& \frac{-5}{16}\end{array}\right]$

$3A=\left[\begin{array}{ccc}-1& 2& 1\\ 1& 0& 0\end{array}\right]+\left[\begin{array}{ccc}\frac{1}{16}& \frac{-2}{16}& \frac{2}{16}\\ \frac{-4}{16}& 0& \frac{-5}{16}\end{array}\right]$

$A=\left[\begin{array}{ccc}\frac{-5}{16}& \frac{5}{8}& \frac{3}{8}\\ \frac{1}{4}& 0& \frac{25}{16}\end{array}\right]$