Question

If A and B are two matrices such n  that AB = B and BA = A , `A^2 + B^2` is equal to

विकल्प

• 2 AB

• 2 BA

• A + B 

•  AB

Answer 1

A + B 

Given: AB = B and  BA = A
\[ A^2 + B^2 = AA + BB\] 
\[ \Rightarrow A^2 + B^2 = BABA + ABAB \left[ \because AB = \text{B and }BA = A \right]\] 

`⇒ A^2 + B^2 = BBA + A AB  ` [∵ AB = B and BA = A ]
\[ \Rightarrow A^2 + B^2 = BA + AB \left[ \because AB =\text{B and }BA = A \right]\] 

\[ \Rightarrow A^2 + B^2 = A + B \left[ \because AB = \text{B and } BA = A \right]\]