If AB = A and BA = B, where A and B are square matrices, then

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B2 = B and A2 = A Here, AB = A . . . \left( 1 ight) BA = B . . . \left( 2 ight) \Rightarrow ABA = AA \left[ 	ext{Multiplying both sides by }A ight] BAB = BB \left[ 	ext{Multiplying both sides by }A ight] \Rightarrow AB = A^2 \left[ 	ext{From eq} . \left( 2 ight) ight] BA = B^2 \left[ 	ext{From eq }. \left( 1 ight) ight] \Rightarrow A = A^2 \left[	ext{ From eq} . \left( 1 ight) ight] B = B^2 \left[ 	ext{From eq} . \left( 2 ight) ight]

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