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Let there be a matrix $$A_n=\left(\begin{matrix}4&2&\cdots&2\\2&4&\ddots&\vdots\\\vdots&\ddots&\ddots&2 \\2&\cdots&2&4\end{matrix}\right)\in M_n\left(\mathbb{Z}_7\right)$$

What is the general formula of $\det\left(A_n\right)$?

decomposition of the matrix to a sum will not help, any suggestions?

anomaly
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gbox
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  • Calculate the case $n=1$ and $n=2$, and then argue by induction that the formula works. You need to use Lagrange's cominor expansion result here to apply induction. Alternatively, you can try row-reducing it, which should not be very difficult here. – Nicolas Bourbaki May 04 '15 at 20:46
  • @NicolasBourbaki can you please send me a link to "Lagrange's cominor expansion" just found "Lagrange Inversion Theorem" – gbox May 04 '15 at 20:48
  • http://en.wikipedia.org/wiki/Determinant#Laplace.27s_formula_and_the_adjugate_matrix

    I made a mistake, it is Laplace, not Lagrange.

     – Nicolas Bourbaki May 04 '15 at 20:50
  • It's $(4-2)^{n-1}(2(n-1)+4)=2^{n-1}(2n+2)=2^n(n+1)$ modulo $7$ (see here) – egreg May 04 '15 at 20:53

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