Question

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.

Answer 1

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.