If p(t) = co + Ct + c2t +...+ Cnt", then define p(A) to be the matrix formed by replacing each power of t in p(t) by the corresponding power of A with A° = I. That is, p(A) = cI+ cA+c2A? +...+ CnA". Show that if A is an eigenvalue of A, then one eigenvalue of p(A) is p(A).

Let B be a 2 × 2 matrix such that 7B²5B+3I = 0. Is B invertible? It so, what are the eigenvalues of B-¹? Justify your answer.

A 2x2 upper triangular matrix A has eigenvalues a and - 5 (with a ≠ -5). An eigenvector corresponding to the eigenvalue a is (1,0). Which of the following could be the matrix A? A. (-5 a/2 0)  B. (a 0/2 -5)  C. (-5 2/0 a)  D. (a -5/0 2)  E. (a 2/0 -5)

In quantum mechanics, when two states with energies ₁ and E2 mix, the resulting states have different energies E and E, which are the eigenvalues of the matrix H = V (EK). V Choosing to be the state with higher energy, derive expressions for E + E, and E - E in terms of E1, E2 and V.