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Prove that A is idempotent if and only if AT is idempotent. Getting Started: The phrase "if and only if" means that you have to prove two statements. 1. If A is idempotent, then AT is idempotent. 2. If AT is idempotent, then A is idempotent. STEP 1: Begin your proof of the first statement by assuming that A is idempotent.

Respuesta :

Answer:

It is proved that A is idempotent if and only if [tex]A^T[/tex] is idempotent.

Step-by-step explanation:

Case-1 : First assume that A is idempotent that is, [tex]A^2=A[/tex]. We have to show by using this [tex](A^T)^2=A^T[/tex]. So,

[tex](A^T)^2=A^T\times A^T=(A\times A)^T=(A^2)^T=A^T[/tex]

Thus [tex]A^T[/tex] is idempotent.

Case-2 :

Conversly, assuming [tex]A^T[/tex] is idempotent, thet is,  [tex](A^T)^2=A^T[/tex] . We have to show [tex]A^2=A[/tex]. So,

[tex]A^2=A\times A=(A^T)^T\times(A^T)^T=(A^T\times A^T)^T[/tex]

[tex]=((A^T)^2)^T=(A^T)^T=A[/tex]

Thus A is idempotent. Hence proved.

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