If A = [1-23-5-60],B=[-1-24215]and C=[24-1-4-36], find the matrix X such that 3A – 4B + 5X = C.

Exercise 2.2 | Q 4 | Page 46 

QUESTION

If A = ${\displaystyle \left[\begin{array}{cc}1& -2\\ 3& -5\\ -6& 0\end{array}\right],\text{B}=\left[\begin{array}{cc}-1& -2\\ 4& 2\\ 1& 5\end{array}\right]\text{and C}=\left[\begin{array}{cc}2& 4\\ -1& -4\\ -3& 6\end{array}\right]}$, find the matrix X such that 3A – 4B + 5X = C.

SOLUTION

3A – 4B + 5X = C
∴ 5X =C + 4B – 3A

= ${\displaystyle \left[\begin{array}{cc}2& 4\\ -1& -4\\ -3& 6\end{array}\right]+4\left[\begin{array}{cc}-1& -2\\ 4& 2\\ 1& 5\end{array}\right]-3\left[\begin{array}{cc}1& -2\\ 3& -5\\ -6& 0\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}2& 4\\ -1& -4\\ -3& 6\end{array}\right]+\left[\begin{array}{cc}-4& -8\\ 16& 8\\ 4& 20\end{array}\right]-\left[\begin{array}{cc}3& -6\\ 9& -15\\ -18& 0\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}2-4- 3& 4-8+6\\ -1+16-9& -4+8+5\\ -3+4+ 18& 6+20-0\end{array}\right]}$

= 5X = ${\displaystyle \left[\begin{array}{cc}-5& 2\\ 6& 19\\ 19& 26\end{array}\right]}$

∴ X = ${\displaystyle \frac{1}{5}\left[\begin{array}{cc}-5& 2\\ 6& 19\\ 19& 26\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}-1& \frac{2}{5}\\ \frac{6}{5}& \frac{19}{5}\\ \frac{19}{5}& \frac{26}{5}\end{array}\right]}$.