Question

(i) Find a matrix X such that 2A + B + X = O, where
A = $\left[\begin{array}{cc}-1& 2\\ 3& 4\end{array}\right]$, B = $\left[\begin{array}{cc}3& -2\\ 1& 5\end{array}\right]$
(ii) If A = $\left[\begin{array}{cc}8& 0\\ 4& -2\\ 3& 6\end{array}\right]$ and B = $\left[\begin{array}{cc}2& -2\\ 4& 2\\ -5& 1\end{array}\right]$, then find the matrix X of order 3 × 2 such that 2A + 3X = 5B.

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right) \\ 2A+B+X=0 \\ \Rightarrow 2\left[\begin{array}{cc}-1& 2\\ 3& 4\end{array}\right]+\left[\begin{array}{cc}3& -2\\ 1& 5\end{array}\right]+X=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}-2& 4\\ 6& 8\end{array}\right]+\left[\begin{array}{cc}3& -2\\ 1& 5\end{array}\right]+X=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}-2+3& 4-2\\ 6+1& 8+5\end{array}\right]+X=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}1& 2\\ 7& 13\end{array}\right]+X=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \therefore X=\left[\begin{array}{cc}-1& -2\\ -7& -13\end{array}\right] \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right) \\ 2A+3X=5B \\ \Rightarrow 2\left[\begin{array}{cc}8& 0\\ 4& -2\\ 3& 6\end{array}\right]+3X=5\left[\begin{array}{cc}2& -2\\ 4& 2\\ -5& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}16& 0\\ 8& -4\\ 6& 12\end{array}\right]+3X=\left[\begin{array}{cc}10& -10\\ 20& 10\\ -25& 5\end{array}\right] \\ \Rightarrow 3X=\left[\begin{array}{cc}10& -10\\ 20& 10\\ -25& 5\end{array}\right]-\left[\begin{array}{cc}16& 0\\ 8& -4\\ 6& 12\end{array}\right] \\ \Rightarrow 3X=\left[\begin{array}{cc}10-16& -10-0\\ 20-8& 10+4\\ -25-6& 5-12\end{array}\right] \\ \Rightarrow 3X=\left[\begin{array}{cc}-6& -10\\ 12& 14\\ -31& -7\end{array}\right] \\ \Rightarrow X=\frac{1}{3}\left[\begin{array}{cc}-6& -10\\ 12& 14\\ -31& -7\end{array}\right] \\ \therefore X=\left[\begin{array}{cc}-2& \frac{-10}{3}\\ 4& \frac{14}{3}\\ \frac{-31}{3}& \frac{-7}{3}\end{array}\right] \end{gathered}$$