Question

Give an example of two matrices A and B whose product is commutative so that$AB=BA$

Short answer

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

Step-by-step solution

Step 1- Determine the dimensions of the required matrices

The dimensions for both the matrices should be same so the dimensions$2\times 2$are the best choice.

– Write the matrices 

Each matrix should contain 0 in place of first row and second column along with the second row and first column so write the matrices.

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

Step 3- Check the matrices

Perform multiplication between the obtained matrices A and B.

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

Now, perform multiplication between B and A.

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

Step 3- Make the conclusion

As the obtained matrices are satisfying the commutative property$AB=BA$so the obtained matrices are the required matrices.