Finding the determinant of given matrix

Asked Apr 04 '20 at 11:47

Active Apr 04 '20 at 11:48

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I’m taking Graph Theory lecture, and in the assignment there is a question about finding the determinant of a matrix $J_n-(1+\lambda)I_n$ where $J_n = (1)_{n \times n}$ and $I_n$ is the identity matrix of order $n$.

I have seen this problem before in the Linear Algebra course, and solved it by computing eigenvalues of the given matrix. However, since the final goal of this problem is finding eigenvalues of a complete graph $K_n$ by using this, it seems to be better to use different method.

I tried to use induction on the size of the matrix, but it does not seem to work well. I would be glad if someone give me a little bit of hint about this question.

edited Apr 04 '20 at 11:48

Rodrigo de Azevedo

asked Apr 04 '20 at 11:47

Alex Lee

In what way would the "complete graph" make the solution different ? – Apr 04 '20 at 12:03
@YvesDaoust It can be solved in a similar way, actually. But I wonder if there might be other way to solve the problem. – Alex Lee Apr 04 '20 at 12:14
Any answer to this question is equivalent to finding the eigenvalues of $J_n$. There are different ways to find these eigenvalues; see this question. – Ben Grossmann Apr 04 '20 at 13:06

Finding the determinant of given matrix

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