10G (source code)
= 10G
{scope}
{title2=Linear Algebra}
Define the <determinant> by the signed permutation sum
$$
\det A=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\prod_{i=1}^n a_{i,\sigma(i)}.
$$
It is multilinear and alternating in the rows, and likewise in the columns. Row multilinearity follows because every product contains exactly one entry from each row; a swap changes the permutation sign, and equal rows give cancellation in pairs. Applying the same argument to the transpose proves the column properties.
Consequently, for either rows or columns, \b[a swap multiplies the determinant by $-1$; multiplication of one row or column by $c$ multiplies it by $c$; addition of a multiple of another row or column leaves it unchanged]. For the last rule, multilinearity separates the original determinant and a determinant with two equal rows or columns, which is zero. These arguments also cover singular matrices and zero scaling.
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