Show that if A is an n x n matrix

  • #1

Homework Statement



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.


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.
 

Answers and Replies

  • #2
35,651
12,217


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


I'm still not understanding what to do with that...
 
  • #4
Dick
Science Advisor
Homework Helper
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Homework Statement



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.


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.
 
  • #5
35,651
12,217


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

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