Chapter 11.4, Problem 4AYU

To determine

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

Answer to Problem 4AYU

The statement “Matrix multiplication is commutative.” is false.

Explanation of Solution

Given information:

The statement “Matrix multiplication is commutative.”

Consider the provided statement “Matrix multiplication is commutative.”

To find the product AB of two matrices A and B, the number of column in matrix A must be equal to number of rows in matrix B.

For example: Let $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$ and $B=\left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right)$ . Number of rows in matrix A are 2 and number of columns are 2. Number rows in matrix B are 2 and number of columns are 2.

Compute the value of $AB$ .

  $\begin{array}{c}AB=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)\left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right)\\ =\left(\begin{array}{cc}5& 4\\ 11& 10\end{array}\right)\end{array}$

Compute the value of $BA$ .

  $\begin{array}{c}BA=\left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right)\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)\\ =\left(\begin{array}{cc}7& 10\\ 5& 8\end{array}\right)\end{array}$

Observe that, $AB\ne BA$ , that is matrices do not follow commutative law of multiplication.

Thus, the statement “Matrix multiplication is commutative.” is false.