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# This equation has multiple correct options;If AB = A and BA = B, then1. ${A^2}B = {A^2}$ 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.

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.
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