Question

Show that matrix addition is commutative; that is, show that if A and B are both m × n matrices, then A + B = B + A.

Short answer

A + B = B + A

Step-by-step solution

Step 1:

Commutative property of addition of real numbers

$a+b=b+a$with$a,b\in R$

Step 2:

A is a$m\times n$matrix and B is a$m\times n$matrix

To proof A + B = B + A

We can rewrite the matrix A as $\left[a_{ij}\right]$where $a_{ij}$represent the element in the ith row and jth column of A

The addition of two matrices add the corresponding elements In the matrices( thus the two elements that are In the same row and column in the two matrices

use the commutative property of addition of real numbers

$\begin{array}{r}A+B=\left[a_{i\ddot{j}}\right]+\left[b_{ij}\right]\\ =\left[a_{ij}+b_{ij}\right]\\ =\left[b_{ij}+a_{ij}\right]\\ =B+A\end{array}$