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  4. If A and B are two square matrices of the same order such that AB = B and BA = A, then A2 + B2 is always equal to ...............

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Q. If $A $ and $ B $ are two square matrices of the same order such that $AB = B$ and $BA = A, $ then $A^2 + B^2$ is always equal to ...............

KCETKCET 2010Matrices

Solution:

Given, $A B=B,\, B A=A$...(i)
Then, $ A^{2}+B^{2}=A \cdot A+B \cdot B$
$=A(B A)+B(A B)$ [from Eq.(i)]
$=(A B) A+(B A) B$ (by commutative law)
$=B A+A B$ [from Eq. (i)]
$=A+B$ [from Eq.(i)]