# A matrix satisfies A^2 - 4A + 5I = 0, then n is even.

by Hydroxide
Tags: determinants
 P: 16 1. The problem statement, all variables and given/known data 1) Let A be an n x n matrix with A^2 -4A +5I = 0. Show that n must be even. 2) Let A be an m x n matrix where m
HW Helper
P: 2,586
 Quote by Hydroxide 1. The problem statement, all variables and given/known data 1) Let A be an n x n matrix with A^2 -4A +5I = 0. Show that n must be even.
An n x n matrix with real entries? If it can have complex entries, this isn't true. Anyways, what do you know about minimal polynomials and characteristic polynomials?
 2) Let A be an m x n matrix where m
What do you know about the relationships between rank, invertibility, and determinants.
 1) (A-2I)^2 +I=0
Okay, that's not bad. So (A-2I)2 = -I. Compute the determinant of both sides.
P: 16
Cheers I've got the first question now. Was easier than i thought.

I still can't do 2) though.

 Quote by AKG What do you know about the relationships between rank, invertibility, and determinants.
 P: 24 It may help to think of matrix multiplication with a vector as a linear combination of the columns of the matrix i.e. For $$A\vec{c} = \vec{b}\\$$ b is a linear combination of the columns of A And hence a matrix multiplication with a vector will produce a matrix whose columns are a linear combination of the columns of the first matrix. i.e. For $$AB = C\\$$ C's columns are linear combinations of the columns of A Sorry if the Latex is less than desirable, as you can see, I'm new here.