Question

If A is diagonalizable then, ____________

(a) A^n = (PDP^-1)^n = PD^nP^n

(b) A^n = (PDP^-1)^n = PD^nP^1

(c) A^n = (PDP^-1)^n = PD^nP^-1

(d) A^n = (PDP^-1)^n = PD^nP

The question was asked during an online interview.

The above asked question is from Diagonalization Powers of a Matrix topic in portion Eigen Values and Eigen Vectors of Engineering Mathematics

Answer 1

Correct answer is (c) A^n = (PDP^-1)^n = PD^nP^-1

The best I can explain: The definition of diagonalization states that, An n × n matrix A is diagonalizable if there exists an n × n invertible matrix P and an n × n diagonal matrix D such that,

P^-1 AP = D

A = PDP^-1

A^n = (PDP^-1)^n = PD^nP^-1