Determinant in $\mathbb Z_{5}$

Question

I need to find

$$ \det\left[ \begin{array}{cc} 2 & 4 & 0 \\ 1 & 1 & 3 \\ 3 & 2 & 1 \end{array} \right] $$ over $\mathbb Z_{5}$

What I did:

$$2\det\left[ \begin{array}{cc} 1 & 3 \\ 2 & 1 \end{array} \right] - 4\det\left[ \begin{array}{cc} 1 & 3 \\ 3 & 1 \end{array} \right]$$

$$=2(1-6) - 4(-8)$$ $$=2(0) - 4(2)$$ $$=2$$

Is this correct? I have never done determinants over a certain set like $\mathbb Z_{5}$ before. Thanks.

Answer 1

The determinant is the same polynomial in the matrix entries no matter which field (or commutative ring) the entries come from.

So what you're doing is right -- you can think of it either as doing the calculations in $\mathbb Z_5$, or as computing the determinant over $\mathbb Z$ and reducing modulo 5 at the very end.