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# If ${\text{A}}$ and ${\text{B}}$ are two matrices such that $AB = B$ and $BA = A$, then ${A^2} + {B^2}$equals.${\text{A}}.$ ${\text{2}}AB$${\text{B}}. {\text{2}}BA$${\text{C}}.$ $A + B$${\text{D}}.$ $AB$

Last updated date: 21st Mar 2023
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Hint:-Here, we go through by writing ${A^2} = A.A$ and ${B^2} = B.B$ then rearrange it.
We have to find ${A^2} + {B^2}$
Given, $AB = B$ and $BA = A$
$\Rightarrow {A^2} = A.A = A\left( {BA} \right) = \left( {AB} \right)A = BA = A$
$\Rightarrow {B^2} = B.B = B.\left( {AB} \right) = \left( {BA} \right)B = AB = B$
$\Rightarrow {A^2} + {B^2} = A + B$
So, option ${\text{C}}$ is the correct answer.