4) Hello please help!!!

Question

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

Answer 1

Since A is diagonalizable, there is an invertible matrix P and a diagonal matrix D so that A = PDP^-1

So, A² = PDP^-1PDP^-1 = PD²P^-1

Since D is a diagonal matrix, so is D².

Therefore, A² is diagonalizable.