1. The problem statement, all variables and given/known data A square matrix A (of some size n x n) satisfies the condition A^2 - 8A + 15I = 0. (a) Show that this matrix is similar to a diagonal matrix. (b) Show that for every positive integer k >= 8 there exists a matrix A satisfying the above condition with tr(A) = k. 2. Relevant equations A^2 - tr(A)A + det(A)I = 0 3. The attempt at a solution I'm not entirely sure what to do but here's the attempt.. I subtracted the given formula from the equation, obtaining (8 - tr(A))A + (15 - det(A))I = 0 So either tr(A) = 8 or A = cI, where c = [15 - det(A)]/tr(A) - 8 So since I've shown that A = cI, c being a constant, is this enough to show that A is similar to a diagonal matrix? For (b), I'm completely lost... If it means use A^2 - kA + 15I = 0 then surely the result will follow straight from the last part? And presumably having k<8 will result in something impossible?