Show that if A is an n x n matrix

[KristenSmith]

Homework Statement

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

Homework Equations

None that I know of.

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.

 

[mfb]

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

 

[KristenSmith]

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

 

[Dick]

Homework Statement

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

Homework Equations

None that I know of.

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.

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.

 

[mfb]

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

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