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We know that the order of $SL_n({\mathbb F}_p)$ is $$p^{n(n-1)/2}(p^n-1)(p^{n-1}-1)\cdots(p^2-1).$$ Dividing by $p^{n^2}$, we deduce the probability that $\det$ takes the value $1$ over $M_n({\mathbb ...