(Solution) - A square matrix A is called idempotent if A2 -(2025 Original AI-Free Solution)

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A square matrix A is called idempotent if A2 = A. (The word idempotent comes from the Latin idem, meaning "same;' and potere, meaning "to have power:' thus, something that is idempotent has the "same power" when squared.)
(a) Find three idempotent 2 × 2 matrices.
(b) Prove that the only invertible idempotent n × n matrix is the identity matrix.