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

SOLUTION

Given equations are

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

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

By (i) x 3 – (ii), we get

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

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

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

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

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

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

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

By (i) – (ii) x 3, we get

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

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

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

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

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

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

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