Determinants

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.

2) Let A be an m x n matrix where m<n. Show that det(A^T x A) = 0

3. The attempt at a solution

1) (A-2I)^2 +I=0

Not sure what to do after this though


Thanks in advance

AKG

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?What do you know about the relationships between rank, invertibility, and determinants.Okay, that's not bad. So (A-2I)² = -I. Compute the determinant of both sides.

Hydroxide

Cheers I've got the first question now. Was easier than i thought.

I still can't do 2) though.

Could you explain further please?

AKG

What are the dimensions of the matrix A^TA? What can you say about the rank of A^TA?

Hydroxide

A^TA is n x n
We haven't covered ranks yet

I know that A^TA can reduced so that it has one row of zero's hence det=0. But I don't know how to show it in general.

Glass

It may help to think of matrix multiplication with a vector as a linear combination of the columns of the matrix

i.e. For [tex]A\vec{c} = \vec{b}\\[/tex] 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 [tex]AB = C\\[/tex] 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.