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Question
If AB = A and BA = B, where A and B are square matrices, then
Options
B2 = B and A2 = A
B2 ≠ B and A2 = A
A2 ≠ A , B2 =B
A2 ≠ A , B2 ≠ B
Solution
B2 = B and A2 = A
\[Here, \]
\[AB = A . . . \left( 1 \right) \]
\[BA = B . . . \left( 2 \right)\]
\[ \Rightarrow ABA = AA \left[ \text{Multiplying both sides by }A \right] \]
\[BAB = BB \left[ \text{Multiplying both sides by }A \right] \]
\[ \Rightarrow AB = A^2 \left[ \text{From eq} . \left( 2 \right) \right] \]
\[BA = B^2 \left[ \text{From eq }. \left( 1 \right) \right]\]
\[ \Rightarrow A = A^2 \left[\text{ From eq} . \left( 1 \right) \right] \]
\[B = B^2 \left[ \text{From eq} . \left( 2 \right) \right]\]
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