Guided Proof Prove that if A2 = A, then either A issingular or A = I. Getting Started

Chapter 2, Problem 70

(choose chapter or problem)

Prove that if \(A^{2} = A\), then either A is singular or A = I.

Getting Started: You must show that either A is singular or A equals the identity matrix.

(i) Begin your proof by observing that A is either singular or nonsingular.

(ii) If A is singular, then you are done.

(iii) If A is nonsingular, then use the inverse matrix \(A^{−1}\) and the hypothesis \(A^{2} = A\) to show that A = I.

Text Transcription:

A^2 = A

A^-1