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[Steps Shown] (a) Suppose that A, B and C are n * n matrices and that C is nonsingular. Prove that B=C^-1 A C implies that A and B have the same eigenvalues

Question: (a) Suppose that A, B and $C$ are $n \times n$ matrices and that $C$ is nonsingular. Prove that $B=C^{-1} A C$ implies that $A$ and $B$ have the same eigenvalues but not vice versa.

(b) Given that $A$ is an $n \times n$ matrix that satisfies $A^{2}=A$, determine its eigenvalues.

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[Steps Shown] (a) Suppose that A, B and C are n * n matrices and that C is nonsingular. Prove that B=C^-1 A C implies that A and B have the same eigenvalues

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