Chapter 3.5, Problem 31HP

To determine

To prove : that matrix addition is commutative for $2\times 2$ matrices.

Explanation of Solution

Given information : Two matrices of order $2\times 2$

Formula used : Addition of real number is commutative.

Proof:

Let $A={\left[a_{i,j}\right]}_{2\times 2}$ and $B={\left[b_{i,j}\right]}_{2\times 2}$ be two matrices.

Now, consider

  $\begin{array}{c}A+B={\left[a_{i,j}\right]}_{2\times 2}+{\left[b_{i,j}\right]}_{2\times 2}\\ ={\left[\left(a_{i,j}+b_{i,j}\right)\right]}_{2\times 2}\\ ={\left[\left(b_{i,j}+a_{i,j}\right)\right]}_{2\times 2}\\ ={\left[b_{i,j}\right]}_{2\times 2}+{\left[a_{i,j}\right]}_{2\times 2}\\ =B+A\end{array}$ Since Addition of real numbers is commutative.

This shows that addition is commutative for $2\times 2$ matrices.