If AB = A and BA = B, then

The correct option is A² = A.

We are given two matrix equations:
1) $AB=A$
2) $BA=B$

We want to find the value of $A^2$ (which is $AA$). Let us start with the definition of $A^2$:

$$A^2=AA$$

Using the first given equation, we can substitute the first $A$ in the product with $AB$:

$$A^2=\left(AB\right)A$$

By the associative property of matrix multiplication, we can regroup the terms as follows:

$$A^2=A\left(BA\right)$$

Now, we substitute the second given equation, $BA=B$, into the expression:

$$A^2=AB$$

Finally, we use the first equation $AB=A$ once again:

$$A^2=A$$

This confirms that the matrix $A$ is an idempotent matrix (i.e., $A^2=A$), which matches the correct option.