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vfiekz [6]
3 years ago
5

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.
Mathematics
1 answer:
Otrada [13]3 years ago
3 0

Answer:

It is proved that A is idempotent if and only if A^T is idempotent.

Step-by-step explanation:

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

(A^T)^2=A^T\times A^T=(A\times A)^T=(A^2)^T=A^T

Thus A^T is idempotent.

Case-2 :

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

A^2=A\times A=(A^T)^T\times(A^T)^T=(A^T\times A^T)^T

=((A^T)^2)^T=(A^T)^T=A

Thus A is idempotent. Hence proved.

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