This equation has multiple correct options;
If AB = A and BA = B, then
1. ${A^2}B = {A^2}$
2. ${B^2}A = {B^2}$
3. $ABA = A$
4. $BAB = B$

Hint: We are going to solve every option from the question and apply the required conditions and operation.

Complete step-by-step answer:
Given, ${AB} = {A}$
Multiply by A
$ \Rightarrow A \cdot AB = A \cdot A \Rightarrow {A^2}B = {A^2}$
So (1) option is correct

Given, ${BA} = {B}$
Multiply by B
\[ \Rightarrow B \cdot BA = B \cdot B \Rightarrow {B^2}A = {B^2}\]
So (2) option is also correct

Given, BA = B
Multiply by A
$ABA = AB$
We know that AB = A
$ \Rightarrow ABA = A$
So (3) option is correct

Take AB = A
Multiply by B
$ \Rightarrow BAB = BA$
We know that BA = B
$ \Rightarrow BAB = B$
So, (4) option is correct
Hence all options are correct.
Note: Such problems you have to use the conditions given and try to apply some operations in given conditions. So you can easily get an answer.