Question

If A = $\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right]$, B = $\left[\begin{array}{ccc}-1& 0& 2\\ 3& 4& 1\end{array}\right]$, C = $\left[\begin{array}{ccc}-1& 2& 3\\ 2& 1& 0\end{array}\right]$, find
(i) A + B and B + C
(ii) 2B + 3A and 3C − 4B.

Answer 1

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

It is not possible to add these matrices because the number of elements in A are not equal to the
number of elements in B. So, A + B does not exist.

$$\begin{gathered} \Rightarrow B+C=\left[\begin{array}{ccc}-1& 0& 2\\ 3& 4& 1\end{array}\right]+\left[\begin{array}{ccc}-1& 2& 3\\ 2& 1& 0\end{array}\right] \\ \Rightarrow B+C=\left[\begin{array}{ccc}-1-1& 0+2& 2+3\\ 3+2& 4+1& 1+0\end{array}\right] \\ \Rightarrow B+C=\left[\begin{array}{ccc}-2& 2& 5\\ 5& 5& 1\end{array}\right] \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right) \\ 2B+3A=2\left[\begin{array}{ccc}-1& 0& 2\\ 3& 4& 1\end{array}\right]+3\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right] \end{gathered}$$

It is not possible to add these matrices because the number of elements in B are not equal to the
number of elements in A. So, 2B + 3A does not exist.

$$\begin{gathered} \Rightarrow 3C-4B=3\left[\begin{array}{ccc}-1& 2& 3\\ 2& 1& 0\end{array}\right]-4\left[\begin{array}{ccc}-1& 0& 2\\ 3& 4& 1\end{array}\right] \\ \Rightarrow 3C-4B=\left[\begin{array}{ccc}-3& 6& 9\\ 6& 3& 0\end{array}\right]-\left[\begin{array}{ccc}-4& 0& 8\\ 12& 16& 4\end{array}\right] \\ \Rightarrow 3C-4B=\left[\begin{array}{ccc}-3+4& 6-0& 9-8\\ 6-12& 3-16& 0-4\end{array}\right] \\ \Rightarrow 3C-4B=\left[\begin{array}{ccc}1& 6& 1\\ -6& -13& -4\end{array}\right] \end{gathered}$$