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Addition and Subtraction of a Matrix

Which of the ...

Question

Which of the following statements is false given AB = A and BA = B where A and B are two $3 \times 3$ matrices?

A^{2} B = A^{2}

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A B A = A

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B A B = B

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A = B

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Solution

The correct option is D $A = B$
Given, $A B = A$ and $B A = B$
We have, $A^{2} B = A (A B) = A A = A^{2}$ $B^{2} A = B (B A) = B B = B^{2}$ $A B A = A (B A) = A B = A$ and $B A B = B (A B) = B A = B .$
But, A and B need not be equal.

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