3R) Please help with this

Question

If A is a diagonalizable n × n matrix, prove that A² is also diagonalizable.

Answer 1

 is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that .

Then P^(-1)A^2P= P^(-1)AAP= P^(-1)APP^(-1)AP=DD=D^2. Since D is diagonal =>D^2 is diagonal too =>A^2 is diagonal.