Chapter 11.4, Problem 2AYU

To determine

Whether the statement “Matrix addition is commutative.” is true or false.

Answer to Problem 2AYU

The statement “Matrix addition is commutative.” is true.

Explanation of Solution

Given information:

The statement “Matrix addition is commutative.”

Consider the provided statement “Matrix addition is commutative.”

If there are two matrices of same order then matrix addition is commutative.

Let A and B be matrices of order $2\times 2$ .

Let $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$ and $B=\left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right)$

Compute $A+B\text{ and }B+A$ .

  $\begin{array}{c}A+B=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)+\left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right)\\ =\left(\begin{array}{cc}2& 4\\ 5& 4\end{array}\right)\end{array}$

And

  $\begin{array}{c}B+A=\left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right)+\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)\\ =\left(\begin{array}{cc}2& 4\\ 5& 4\end{array}\right)\end{array}$

Observe that, $A+B=B+A$ , that is matrices follow commutative law of addition provided both matrices have same order.

Thus, the statement “Matrix addition is commutative.” is true.