Question

A tricky question, A is a real 2x2 matrix with distinct eigenvalues, A^2 has two distinct eigenvalues, true or false?

Answer 1

false as eigen values of A^2 are same as square of eigen values of A
consider eigen values of A that are additive inverses of each other : e, -e

Answer 2

The way I did it, was to assume the eigenvalues were a and -a for A and this implies it's diagonalizable. \[\Large A^2 = PD^2P^{-1}\] and so we know it is false because there are repeated eigenvalues now, both a^2.