Chapter 2.4, Problem 54E

Guided Proof Prove that $A$ is idempotent if and only if $A^T$ is idempotent.

Getting Started: The phrase “if and only if” means that you have to prove two statements:

1.If $A$ is idempotent, then $A^T$ is idempotent.

2.If $A^T$ is idempotent, then $A$ is idempotent.

(i) Begin your proof of the first statement by assuming that $A$ is idempotent.

(ii) This means that ${ A}^2=A$.

(iii) Use the properties of the transpose to show that ${ A}^T$ is idempotent.

(iv) Begin your proof of the second statement by assuming that $A^T$ is idempotent.