If AB = A and BA = B, then which of the following is/are true? - Sri Chaitanya Infinity Learn Best Online Courses for NCERT Solutions, CBSE, ICSE, JEE, NEET, Olympiad and Class 6 to 12

Given, AB = A, BA = B

$\Rightarrow B \times AB = B \times A$

or (BA)B = B or B²= B

Also,

$\begin{aligned} A \times BA = AB \\ \Rightarrow (AB) A = A \\ \Rightarrow A^{2} = A \end{aligned}$

Now ${(A^{T})}^{2} = (A^{T} \times A^{T}) = (A \times A)^{T} = {(A^{2})}^{T} = A^{T}$

Similarly, (B^T)² = B^T

Hence, A^T and B^T are idempotent.

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