Chapter 3.6, Problem 48HP

To determine

To write : two matrices A and B such that $AB=BA$ .

Answer to Problem 48HP

The required matrices are $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 4\\ 3& 1\end{array}\right]$.

Explanation of Solution

Given information : Find two matrices A and B.

Since, matrix multiplication is not commutative in general.

Now, let

  $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 4\\ 3& 1\end{array}\right]$

We see that

  $\begin{array}{c}AB=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}2& 4\\ 3& 1\end{array}\right]\\ =\left[\begin{array}{cc}2& 4\\ 3& 1\end{array}\right]\end{array}$

And

  $\begin{array}{c}BA=\left[\begin{array}{cc}2& 4\\ 3& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\\ =\left[\begin{array}{cc}2& 4\\ 3& 1\end{array}\right]\end{array}$

Here the multiplication is commutative.

i.e AB=BA