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

[Hydroxide]

Homework Statement

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

The Attempt at a Solution

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

Not sure what to do after this though


Thanks in advance

 

[AKG]

Homework Statement

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<n. Show that det(A^T x A) = 0

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.

 

[Hydroxide]

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

I still can't do 2) though.

What do you know about the relationships between rank, invertibility, and determinants.

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]

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

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 $$A\vec{c} = \vec{b}\\$$ 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.