4 Multiplication of a matrix by a number

There is also a natural way of defining the product of a matrix with a number. Using the matrix $A$ above, we note that

$A+A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right]+\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 4\hfill \\ \hfill 6\hfill & \hfill 8\hfill \end{array}\right]$

What we see is that $2A$ (which is the shorthand notation for $A+A$ ) is obtained by multiplying every element of $A$ by $2$ .

In general if $A$ is an $m\times n$ matrix with typical element $a_{ij}$ then the product of a number $k$ with $A$ is written $kA$ and has the corresponding elements $k{a}_{ij}$ .

Hence, again using the matrix $A$ above,

$7A=7\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 7& \hfill 14\\ \hfill 21& \hfill 28\end{array}\right]$

Similarly:

$-3A=\left[\begin{array}{cc}\hfill -3& \hfill -6\\ \hfill -9& \hfill -12\end{array}\right]$

Task!

For the following matrices find, where possible, $A+B,A-B,B-A,2A$ .

1. $A=\left[\begin{array}{cc}\hfill 1& \hfill 2\\ \hfill 3& \hfill 4\end{array}\right]B=\left[\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill 1& \hfill 1\end{array}\right]$
2. $A=\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 3\\ \hfill 4& \hfill 5& \hfill 6\\ \hfill 7& \hfill 8& \hfill 9\end{array}\right]B=\left[\begin{array}{ccc}\hfill 1& \hfill 1& \hfill 1\\ \hfill -1& \hfill -1& \hfill -1\\ \hfill 1& \hfill 1& \hfill 1\end{array}\right]$
3. $A=\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 3\\ \hfill 4& \hfill 5& \hfill 6\\ \hfill 7& \hfill 8& \hfill 9\end{array}\right]B=\left[\begin{array}{cc}\hfill 1& \hfill 2\\ \hfill 3& \hfill 4\\ \hfill 5& \hfill 6\end{array}\right]$

Answer