3 Addition and subtraction of matrices

Under what circumstances can we add two matrices i.e. define $A+B$ for given matrices $A$ , $B$ ?

Consider

$A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right]\text{and}B=\left[\begin{array}{ccc}\hfill 5& \hfill 6& \hfill 9\\ \hfill 7& \hfill 8& \hfill 10\end{array}\right]$

There is no sensible way to define $A+B$ in this case since $A$ and $B$ are different sizes.

However, if we consider matrices of the same size then addition can be defined in a very natural way. Consider $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right]$ and $B=\left[\begin{array}{cc}\hfill 5\hfill & \hfill 6\hfill \\ \hfill 7\hfill & \hfill 8\hfill \end{array}\right]$ . The ‘natural’ way to add $A$ and $B$ is to add corresponding elements together:

$A+B=\left[\begin{array}{cc}\hfill 1+5\hfill & \hfill 2+6\hfill \\ \hfill 3+7\hfill & \hfill 4+8\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 6& \hfill 8\\ \hfill 10& \hfill 12\end{array}\right]$

In general if $A$ and $B$ are both $m\times n$ matrices, with elements $a_{ij}$ and $b_{ij}$ respectively, then their sum is a matrix $C$ , also $m\times n$ , such that the elements of $C$ are

$c_{ij}=a_{ij}+b_{ij}i=1,2,\dots ,mj=1,2,\dots ,n$

In the above example

$c_{11}=a_{11}+b_{11}=1+5=6c_{21}=a_{21}+b_{21}=3+7=10\text{and so on.}$

Subtraction of matrices follows along similar lines:

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