Solve the following equations for X and Y, if 3X − Y = [1-1-11] and X – 3Y = [0-10-1].

Exercise 2.2 | Q 11 | Page 47

QUESTION

Solve the following equations for X and Y, if 3X − Y = $\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$  and X – 3Y = $\left[\begin{array}{cc}0& -1\\ 0& -1\end{array}\right]$.

SOLUTION

Given equations are

3X – Y = $\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$          ...(i)

and X – 3Y = $\left[\begin{array}{cc}0& -1\\ 0& -1\end{array}\right]$     ...(ii)

By (i) x 3 – (ii), we get

8X = $3\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]-\left[\begin{array}{cc}0& -1\\ 0& -1\end{array}\right]$

$3\left[\begin{array}{cc}3& -3\\ -3& 3\end{array}\right]-\left[\begin{array}{cc}0& -1\\ 0& -1\end{array}\right]$

$\left[\begin{array}{cc}3-0& -3+1\\ -3-0& 3+1\end{array}\right]$

∴ 8X = $\left[\begin{array}{cc}3& -2\\ -3& 4\end{array}\right]$

∴ X = $\frac{1}{8}\left[\begin{array}{cc}3& -2\\ -3& 4\end{array}\right]$

∴ X = $\left[\begin{array}{cc}\frac{3}{8}& \frac{-2}{8}\\ \frac{-3}{8}& \frac{4}{8}\end{array}\right]$

$\left[\begin{array}{cc}\frac{3}{8}& \frac{-1}{4}\\ \frac{-3}{8}& \frac{1}{2}\end{array}\right]$

By (i) – (ii) x 3, we get

8Y = $\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]-3\left[\begin{array}{cc}0& -1\\ 0& -1\end{array}\right]$

$\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]-\left[\begin{array}{cc}0& -3\\ 0& -3\end{array}\right]$

$\left[\begin{array}{cc}1-0& -1+3\\ -1-0& 1+3\end{array}\right]$

∴ 8Y = $\left[\begin{array}{cc}1& 2\\ -11& 4\end{array}\right]$

∴ Y = $\frac{1}{8}\left[\begin{array}{cc}1& 2\\ -11& 4\end{array}\right]$

$\left[\begin{array}{cc}\frac{1}{8}& \frac{2}{8}\\ \frac{-1}{8}& \frac{4}{8}\end{array}\right]$

$\left[\begin{array}{cc}\frac{1}{8}& \frac{1}{4}\\ \frac{-1}{8}& \frac{1}{2}\end{array}\right]$.