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Define the determinant by the signed permutation sumIt 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, 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.