I only need (c) and (d) and you will get full points! Problem 3. Given an n x n

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I only need (c) and (d) and you will get full points!

Problem 3. Given an n x n matrix A and a polynomial p(x) = c0 + c1x + ... + ckx^4, we define p(A) as the matrix c0In + c1A + ... + ckA^k. (c) Let A be any n x n matrix. Prove that there exists k element of N and a unique polynomial mA(x) with leading coefficient 1 and of least degree k such that mA(A) = 0. (The polynomial mA(x) is called the minimal polynomial of A.) Hint : Show that there exists k such that I, A, A^2, . . . , Ak element of Rnxn are linearly dependent. d) Prove that if lambda is an eigenvalue of A, then mA(lambda) = 0

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