First slide

Operations of matrix

Question

If AB = A and BA = B, then which of the following is/are true?

Moderate

A

A is idempotent
B

B is idempotent
C

A^T is idempotent
D

none of these

Solution

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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