Q100 / 180 · COMEDK 2014 · Mathematics

$A$ and $B$ are two matrices such that $A B=B$, $B A=A$, then $A^{2}+B^{2}=$

  • A
    $2 A B$
  • B
    $2 B A$
  • Correct answer
    C
    $A+B$
  • D
    $A B$

Solution

Given, $A B=B$ Multiply by $A$ on both sides $$ A B A=B A \quad \text{...(i)} $$ Also, $B A=A$ Multiply by $B$ on both sides $$ B A B=A B \quad \text{...(ii)} $$ Adding Eqs. (i) and (ii), we get $A B A+B A B=B A+A B$ $\Rightarrow \quad A(B A)+B(A B)=B A+A B$ $\Rightarrow \quad A A+B B=A+B$ $\Rightarrow \quad A^{2}+B^{2}=A+B$

Hence, the correct answer is option: 'C'.

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