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Question: If \({\text{A}}\) and \({\text{B}}\) are two matrices such that \(AB = B\) and \(BA = A\), then \({A...

If A{\text{A}} and B{\text{B}} are two matrices such that AB=BAB = B and BA=ABA = A, then A2+B2{A^2} + {B^2}equals.
A.{\text{A}}. 2AB{\text{2}}AB
B.{\text{B}}. 2BA{\text{2}}BA
C.{\text{C}}. A+BA + B
D.{\text{D}}. ABAB

Explanation

Solution

Hint:-Here, we go through by writing A2=A.A{A^2} = A.A and B2=B.B{B^2} = B.B then rearrange it.

We have to find A2+B2{A^2} + {B^2}
Given, AB=BAB = B and BA=ABA = A
⇒A2=A.A=A(BA)=(AB)A=BA=A\Rightarrow {A^2} = A.A = A\left( {BA} \right) = \left( {AB} \right)A = BA = A
⇒B2=B.B=B.(AB)=(BA)B=AB=B\Rightarrow {B^2} = B.B = B.\left( {AB} \right) = \left( {BA} \right)B = AB = B
⇒A2+B2=A+B\Rightarrow {A^2} + {B^2} = A + B
So, option C{\text{C}} is the correct answer.

Note:-Whenever we face such a type of question of matrix the key concept for solving the question is you have to proceed according to what is given in question, and try to rearrange the terms to get an answer.