September 16th, 2013, 09:29 AM  #1 
Newbie Joined: Sep 2013 Posts: 2 Thanks: 0  Eigenvalue Problem
If A is the 4 by 4 matrix of ones, what are the eigenvalues and the determinant of A  I? How to solve this without expanding?

September 16th, 2013, 10:15 AM  #2 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: Eigenvalue Problem
The determinant will be 0, as you will have 1 1 + 1  1 times the determinant of the same 3x3 matrix with all 1's. Any vector (a, b, c, d) will become (a+b+c+d, a+b+c+d, a+b+c+d, a+b+c+d) under the action of this matrix. If this is an eigenvector with eigenvalue k, then a+b+c+d = ka = kb =kc = kd. Hence a = b = c = d and k = 4. 
September 16th, 2013, 12:16 PM  #3 
Newbie Joined: Sep 2013 Posts: 2 Thanks: 0  Re: Eigenvalue Problem
I think you wrote about the eigenvalues and determinant of the matrix A. But I need those of the matrix AI. I understand A has eigenvalues 4,0,0,0 and Determinant is 0. But how do I get the results for AI from this?

September 16th, 2013, 01:00 PM  #4 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: Eigenvalue Problem
The matrix A  I acting on the vector (a, b, c, d) will produce the vector (b+c+d, a+c+d, a+b+d, a+b+c). If this is an eigenvector then either: a = b = c =d and the eigenvalue k = 3 or k = 1 and a = (b+c+d) This can be seen by solving the system of simultaneous equations. The determinant of A is 9. 

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