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a few things to possibly add

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a few things to possibly add

* $|SL(n,q)| =
\frac{1}{q-1}\prod_{i=0}^{n-1}(q^n-q^i)$ where q is the order of the field.

* $[SL(n,q),SL(n,q)] = SL(n,K)$ with two exceptions:
$SL(2,2)$ and $SL(2,3)$ where q is the order of the field.

Parting words from the person who closed the correction: 
Just like for GL, I think these results would be great to add into a "$\operatorname{SL}(n,\mathbb{F}_q)$" entry, but are of too narrow interest for SL in general.
Status: Rejected
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The determinant of a matrix A⁒(t)=Ai⁒j⁒(t)𝐴𝑑subscript𝐴𝑖𝑗𝑑 is given by

d⁒e⁒t⁒(Ai⁒j⁒(t))=ei⁒j⁒k⁒Ai⁒1⁒(t)⁒Aj⁒2⁒(t)⁒Ak⁒3⁒(t),𝑑𝑒𝑑subscript𝐴𝑖𝑗𝑑subscriptπ‘’π‘–π‘—π‘˜subscript𝐴𝑖1𝑑subscript𝐴𝑗2𝑑subscriptπ΄π‘˜3𝑑

where ei⁒j⁒ksubscriptπ‘’π‘–π‘—π‘˜ is the Cartesian alternator. Let us take the derivative of this expression with respect to the parameter t𝑑,

dd⁒t⁒d⁒e⁒t⁒(Ai⁒j⁒(t))=ei⁒j⁒k⁒(Ai⁒1⁒(t),t⁒Aj⁒2⁒(t)⁒Ak⁒3⁒(t)+Ai⁒1⁒(t)⁒Aj⁒2⁒(t),t⁒Ak⁒3⁒(t)+Ai⁒1⁒(t)⁒Aj⁒2⁒(t)⁒Ak⁒3⁒(t),t),𝑑𝑑𝑑𝑑𝑒𝑑subscript𝐴𝑖𝑗𝑑subscriptπ‘’π‘–π‘—π‘˜subscript𝐴𝑖1𝑑𝑑subscript𝐴𝑗2𝑑subscriptπ΄π‘˜3𝑑subscript𝐴𝑖1𝑑subscript𝐴𝑗2𝑑𝑑subscriptπ΄π‘˜3𝑑subscript𝐴𝑖1𝑑subscript𝐴𝑗2𝑑subscriptπ΄π‘˜3𝑑𝑑

where β€œ,tfragments,t” stands for derivation. Let us now consider the quantity

Ck⁒3⁒(t)=ei⁒j⁒k⁒Ai⁒1⁒(t)⁒Aj⁒2⁒(t)=12⁒(ei⁒j⁒k⁒Ai⁒1⁒(t)⁒Aj⁒2⁒(t)+ej⁒i⁒k⁒Aj⁒1⁒(t)⁒Ai⁒2⁒(t))=12⁒ei⁒j⁒k⁒(Ai⁒1⁒(t)⁒Aj⁒2⁒(t)-Ai⁒2⁒(t)⁒Aj⁒1⁒(t)),subscriptπΆπ‘˜3𝑑subscriptπ‘’π‘–π‘—π‘˜subscript𝐴𝑖1𝑑subscript𝐴𝑗2𝑑12subscriptπ‘’π‘–π‘—π‘˜subscript𝐴𝑖1𝑑subscript𝐴𝑗2𝑑subscriptπ‘’π‘—π‘–π‘˜subscript𝐴𝑗1𝑑subscript𝐴𝑖2𝑑12subscriptπ‘’π‘–π‘—π‘˜subscript𝐴𝑖1𝑑subscript𝐴𝑗2𝑑subscript𝐴𝑖2𝑑subscript𝐴𝑗1𝑑

which is the same as

Ck⁒3⁒(t)=12⁒ei⁒j⁒k⁒el⁒m⁒3⁒Ai⁒l⁒(t)⁒Aj⁒m⁒(t),subscriptπΆπ‘˜3𝑑12subscriptπ‘’π‘–π‘—π‘˜subscriptπ‘’π‘™π‘š3subscript𝐴𝑖𝑙𝑑subscriptπ΄π‘—π‘šπ‘‘

and in general,

Ck⁒n⁒(t)=12⁒ei⁒j⁒k⁒el⁒m⁒n⁒Ai⁒l⁒(t)⁒Aj⁒m⁒(t).subscriptπΆπ‘˜π‘›π‘‘12subscriptπ‘’π‘–π‘—π‘˜subscriptπ‘’π‘™π‘šπ‘›subscript𝐴𝑖𝑙𝑑subscriptπ΄π‘—π‘šπ‘‘

This is the so-called β€œadjugate” a⁒d⁒jπ‘Žπ‘‘π‘— of the matrix A⁒(t)𝐴𝑑 (notice that this formula is also useful to find the inverse of A⁒(t)𝐴𝑑 whenever it be no singular). Thus, the sum for dd⁒t⁒d⁒e⁒t⁒(Ai⁒j⁒(t))𝑑𝑑𝑑𝑑𝑒𝑑subscript𝐴𝑖𝑗𝑑 above, may be expressed by

dd⁒t⁒d⁒e⁒t⁒(Ai⁒j⁒(t))=12⁒ei⁒j⁒k⁒el⁒m⁒n⁒Ai⁒l⁒(t)⁒Aj⁒m⁒(t)⁒Ak⁒n⁒(t),t=Ck⁒n⁒(t)⁒Ak⁒n⁒(t),t,formulae-sequence𝑑𝑑𝑑𝑑𝑒𝑑subscript𝐴𝑖𝑗𝑑12subscriptπ‘’π‘–π‘—π‘˜subscriptπ‘’π‘™π‘šπ‘›subscript𝐴𝑖𝑙𝑑subscriptπ΄π‘—π‘šπ‘‘subscriptπ΄π‘˜π‘›π‘‘π‘‘subscriptπΆπ‘˜π‘›π‘‘subscriptπ΄π‘˜π‘›π‘‘π‘‘

which is clearly the β€œtrace” t⁒rπ‘‘π‘Ÿ of this product. So that, in matrix notation we have

d⁒(d⁒e⁒t⁒A⁒(t))d⁒t=t⁒r⁒(a⁒d⁒j⁒(A⁒(t))⁒d⁒(A⁒(t))d⁒t).π‘‘π‘‘π‘’π‘‘π΄π‘‘π‘‘π‘‘π‘‘π‘Ÿπ‘Žπ‘‘π‘—π΄π‘‘π‘‘π΄π‘‘π‘‘π‘‘

Indeed this formula is also valid for any finite dimension n𝑛, and its proof is a little more elaborated. We have exposed here the case for n=3𝑛3 for reason of brevity. Also note that Ck⁒nT=Cn⁒ksuperscriptsubscriptπΆπ‘˜π‘›π‘‡subscriptπΆπ‘›π‘˜ is a cofactor of the matrix A⁒(t).𝐴𝑑

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