# Show that if A is an n x n matrix

1. Dec 11, 2012

### KristenSmith

1. The problem statement, all variables and given/known data

Show that if A is an n x n matrix whose kth row is the same as the kth row of In, then 1 is an eigenvalue of A.

2. Relevant equations None that I know of.

3. The attempt at a solution I tried creating an arbitrary matrix A and set it up to to find the det(A-λI) but that got me no where.

2. Dec 11, 2012

### Staff: Mentor

Re: Eigenvalues

If you know that A and At have the same eigenvalues, you can work with the original definition of eigenvalue: There is a vector x such that Atx = 1x.

3. Dec 11, 2012

### KristenSmith

Re: Eigenvalues

I'm still not understanding what to do with that...

4. Dec 11, 2012

### Dick

Re: Eigenvalues

Think about an expansion by minors along the kth row of A-λI. You want to show 1-λ is a factor of the characteristic polynomial.

5. Dec 11, 2012

### Staff: Mentor

Re: Eigenvalues

It is easy to find a specific vector x with Atx = 1x. That shows that x is an eigenvector of At with eigenvalue 1 => At has 1 as eigenvalue => A has 1 as eigenvalue.