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How to calculate the following determinants
Computing determinant of a specific matrix.
How can one compute the determinant of an $n\times n$ matrix where all the diagonal entries are equal to $0$ and all the off-diagonal entries are equal to $1$?
Let $A$ denote the $n\times n$ matrix of all ones. The rank of $A$ is 1, and it can be seen that the vector of all ones is an eigenvector for $A$ with eigenvalue $n$, so the characteristic polynomial of $A$ is $t^{n-1}(t-n)$. If $B$ denotes your matrix, then \begin{eqnarray*} \det(B)&=&\det(A-1)\\ &=&(-1)^n\det(1-A)\\ &=&(-1)^n(t^{n-1}(t-n))|_{t=1}\\ &=&(-1)^n(1-n) \end{eqnarray*}
