Question

Evaluate the following:
(i) $\left(\left[\begin{array}{cc}1& 3\\ -1& -4\end{array}\right]+\left[\begin{array}{cc}3& -2\\ -1& 1\end{array}\right]\right)\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 6\end{array}\right]$

(ii) $[123]\left[\begin{array}{ccc}1& 0& 2\\ 2& 0& 1\\ 0& 1& 2\end{array}\right]\left[\begin{array}{c}2\\ 4\\ 6\end{array}\right]$

(iii) $\left[\begin{array}{cc}1& -1\\ 0& 2\\ 2& 3\end{array}\right]\left(\left[\begin{array}{ccc}1& 0& 2\\ 2& 0& 1\end{array}\right]-\left[\begin{array}{ccc}0& 1& 2\\ 1& 0& 2\end{array}\right]\right)$

Answer 1

$$\begin{gathered} \left(i\right)\left(\left[\begin{array}{cc}1& 3\\ -1& -4\end{array}\right]+\left[\begin{array}{cc}3& -2\\ -1& 1\end{array}\right]\right)\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 6\end{array}\right] \\ \Rightarrow \left(\left[\begin{array}{cc}1+3& 3-2\\ -1-1& -4+1\end{array}\right]\right)\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 6\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}4& 1\\ -2& -3\end{array}\right]\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 6\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}4+2& 12+4& 20+6\\ -2-6& -6-12& -10-18\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}6& 16& 26\\ -8& -18& -28\end{array}\right] \end{gathered}$$


$$\begin{gathered} \left(ii\right)\left[\begin{array}{ccc}1& 2& 3\end{array}\right]\left[\begin{array}{ccc}1& 0& 2\\ 2& 0& 1\\ 0& 1& 2\end{array}\right]\left[\begin{array}{c}2\\ 4\\ 6\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}1+4+0& 0+0+3& 2+2+6\end{array}\right]\left[\begin{array}{c}2\\ 4\\ 6\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}5& 3& 10\end{array}\right]\left[\begin{array}{c}2\\ 4\\ 6\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}10+12+60\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}82\end{array}\right] \end{gathered}$$

$$\begin{gathered} \left(iii\right)\left[\begin{array}{cc}1& -1\\ 0& 2\\ 2& 3\end{array}\right]\left(\left[\begin{array}{ccc}1& 0& 2\\ 2& 0& 1\end{array}\right]-\left[\begin{array}{ccc}0& 1& 2\\ 1& 0& 2\end{array}\right]\right) \\ \Rightarrow \left[\begin{array}{cc}1& -1\\ 0& 2\\ 2& 3\end{array}\right]\left[\begin{array}{ccc}1-0& 0-1& 2-2\\ 2-1& 0-0& 1-2\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}1& -1\\ 0& 2\\ 2& 3\end{array}\right]\left[\begin{array}{ccc}1& -1& 0\\ 1& 0& -1\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}1-1& -1+0& 0+1\\ 0+2& 0+0& 0-2\\ 2+3& -2+0& 0-3\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}0& -1& 1\\ 2& 0& -2\\ 5& -2& -3\end{array}\right] \end{gathered}$$