# 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. $BAB = B$

Answer

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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.

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.

Last updated date: 25th Sep 2023

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