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Find the determinant of the nbyn matrix of p as its diagonal entries and 1 else where.

I tried to solve the problem via two methods, to no avail. First was reducing the matrix by elementary row operations to an 'almost diagonal matrix looking a bit like the jordan form but the first row cant be eliminated posing aproblem.

second I tried to use the definition of determinant by p^x by choosing x number of p from n, then finding how many ways are there to permute the rest of the ones, but I cant find a way to resolve the signs.

Jan Lynn
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1 Answers1

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Find the characteristic polynomial of a matrix with $-1$ in every entry. This is by definition $$\mathrm{det} \begin{pmatrix} t+1 & 1 & ... & 1 \\ 1 & t+1 & ... & 1 \\ .. & .. & .. & .. \\ 1 & 1 & .. & t+1 \end{pmatrix}$$ but also, the matrix with $-1$ in each entry clearly has rank one (so it has eigenvalue $0$ with multiplicity at least $n-1$) and the sum of the eigenvalues is the trace $-n$, so the characteristic polynomial is $t^{n-1}(t+n)$. Now set $t = p-1$ and you're done.

Cocopuffs
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