Properties of the determinant

Computing a large determinant by cofactors is painfully slow: the number of terms grows like \(n!\). The way out is to notice what each elementary row operation does to the determinant and exploit it: reduce the matrix to triangular form, keep track of the changes, and multiply the diagonal at the end. The properties are therefore not loose curiosities but an algorithm in disguise.

There are three basic rules, one per elementary operation. Swapping two rows changes the sign; multiplying a row by \(k\) multiplies the determinant by \(k\); adding a multiple of one row to another leaves it unchanged. From these it follows that two equal or proportional rows give \(0\), because subtracting them produces a zero row. The rules involving the whole matrix, \(\det(AB)=\det A\,\det B\) and \(\det A^T=\det A\), explain why everything said about rows also holds for columns.

Guiding example: if \(A\) is \(3\times3\) with \(\det A=2\), then \(\det(3A)=3^3\cdot2=54\), not \(6\), because each of the three rows is multiplied by \(3\). Two typical mistakes: believing \(\det(A+B)=\det A+\det B\) (false in general; the determinant is linear in each row separately, not in the whole matrix) and forgetting the sign change when swapping rows during reduction. These properties prepare Cramer's rule and the invertibility criterion: \(\det A\neq0\) if and only if \(A\) is invertible.