Given this 4x4 matrix, is there an easier way to calculate the eigenvalues?

Question

This came up as a textbook question: Find the rank and 4 eigenvalues of A, where A is the 4x4 matrix with all 1 entries

1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1

I am trying to calculate the eigenvalues using the determinant of A-\I, but it is proving quite tedious using the cofactor formula. This chapter mentioned trace and product of eigenvalues, and I can't help but think there should be an easier way for me to compute the eigenvalues of this matrix. Is there? You don't have to actually solve the problem for me, just let me know if there's an easier way I could be doing this.. thanks.

Answer 1

This is a rank one matrix, so the nullspace has dimension $3$. This gives you one eigenvalue (which one?) with multiplicity $3$. Then it remains to find the last eigenvalue (try multiplying the matrix with a vector of all $1$s).