Show that if A is an n x n matrix

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Homework Help Overview

The problem involves an n x n matrix A, specifically focusing on the condition that its kth row matches the kth row of the identity matrix In. The objective is to demonstrate that 1 is an eigenvalue of A.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the definition of eigenvalues and consider the relationship between A and its transpose At. Some explore the characteristic polynomial and the implications of the determinant of (A - λI). Others express confusion about how to proceed with the problem.

Discussion Status

There are various lines of reasoning being explored, including the use of specific vectors and the properties of eigenvalues related to transposes. Some participants have suggested methods to approach the problem, but there is no clear consensus on the next steps or a definitive method being followed.

Contextual Notes

Participants note the absence of specific equations or methods that could guide their attempts. The discussion reflects uncertainty about how to leverage the given conditions effectively.

KristenSmith
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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.
 
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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.
 


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


KristenSmith said:

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.
 


KristenSmith said:
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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