1. The problem statement, all variables and given/known data a.) If A is an 'n x n' matrix and X is an 'n x 1' nonzero column matrix with AX = 0 show, by assuming the contrary, that det(A) = 0 b.) Using the answer in 'a' show that the scalar equation which gives the values of λ that satisfy the matrix equation AX = λIX is: det(A - λI) = 0 2. The attempt at a solution a.) If det(A) ≠ 0 then A^-1 exists. X = A^-1 x (AX) = A^-1 x (0) = 0 This is a contradiction because x nonzero so det(A) = 0.... this bit I understand however the next part b.) AX = λIX -> X(A - λI) = 0 X ≠ 0 so for the equation to be true A - λI = 0 I'm not sure how to apply the first result to the second question? It's from Bimore & Davies Calculus: Concepts and Methods book. Any help would be appreciated!