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Question
If A and B are two matrices such n that AB = B and BA = A , `A^2 + B^2` is equal to
Options
2 AB
2 BA
A + B
AB
Solution
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 ]
\[ \Rightarrow A^2 + B^2 = BA + AB \left[ \because AB =\text{B and }BA = A \right]\]
\[ \Rightarrow A^2 + B^2 = A + B \left[ \because AB = \text{B and } BA = A \right]\]
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