Question

If A and B are invertible matrices, then AB must be similar to BA.

Short answer

The above statement is false.

If A and B are two invertible matrices, then AB may not be similar to BA.

Step-by-step solution

Definition of similar matrix

Let A and B are two square matrices, the matrix A is said to be similar to matrix B if there exists an invertible matrix P such that

$B=P^{-1}AP$

Mentioning concept

Since it is given that matrix A and matrix B are invertible, then$A^{-1}\text{and}{B}^{-1}$ exist and are also invertible.

If we multiply the matrix AB by $A^{-1}$from left and from right, we get the matrix

$$\begin{gathered} A^{-1}\left(AB\right)A=\left(A^{-1}A\right)\left(BA\right) \\ \Rightarrow A^{-1}\left(AB\right)A=BA\text{(}\because A^{-1}A=I\text{)} \end{gathered}$$

This shows that matrix BA is similar to matrix AB.

Now, if we multiply the matrix BA by $B^{-1}$from left and from right, we get the matrix

$$\begin{gathered} B^{-1}\left(BA\right)B=\left(B^{-1}B\right)\left(AB\right) \\ \Rightarrow B^{-1}\left(BA\right)B=AB\text{(}\because B^{-1}B=I\text{)} \end{gathered}$$

This shows that matrix AB is similar to matrix BA.

Final Answer

If A is an invertible matrix then BA is similar to AB and if B is an invertible matrix then AB is similar to BA.