If A and B are two matrices such that AB = B

Question:

If $A$ and $B$ are two matrices such that $A B=B$ and $B A=A, A^{2}+B^{2}$ is equal to

(a) $2 A B$

(b) $2 B A$

(c) $A+B$

(d) $A B$

Solution:

(c) $A+B$

Given : $A B=B$ and $B A=A$

$A^{2}+B^{2}=A A+B B$

$\Rightarrow A^{2}+B^{2}=B A B A+A B A B \quad[\because A B=B$ and $B A=A]$

$\Rightarrow A^{2}+B^{2}=B B A+A A B \quad[\because A B=B$ and $B A=A]$

$\Rightarrow A^{2}+B^{2}=B A+A B \quad[\because A B=B$ and $B A=A]$

$\Rightarrow A^{2}+B^{2}=A+B \quad[\because A B=B$ and $B A=A]$

 

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