Let $A$ be a matrix with no repeated eigenvalues: $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}.$ Let $p(x)$ and $r(x)$ be two polynomials satisfying $$p(\lambda_{i})=r(\lambda_{i}) \text{ for } i = 1, 2, \ldots, n.$$ Show that $p(A)=r(A).$
Show matrix polynomials are equal
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linear-algebra
matrices