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A determinantal function on $M_n(F)$ is an alternating multilinear map of the $n$ columns, normalized to take value one at the identity matrix. It is uniquely the determinant. Indeed, expand each column in the standard basis. Terms with a repeated basis vector vanish, and all remaining terms differ from the value on the standard basis only by the sign of a permutation. Thus an alternating multilinear map of $n$ columns is determined by its value on the identity matrix.

We have $D(I_n)=\det A$. Define a function of the columns of $B$ byIt is an alternating multilinear map, because both summands are, and $F(I_n)=\det A+1-\det A=1$. Hence $F$ is a normalized determinantal function. By the asserted uniqueness of the determinant, $F(B)=\det B$ for every $B$. Rearranging yieldsThis argument also covers $\det A=0$ without dividing by it.