Properties of Determinants

Evaluating every determinant by direct expansion is reliable but slow. A standard set of properties, all provable from the expansion rule itself, lets many determinants be simplified — or shown to be zero — with far less arithmetic. These properties are tested both as a topic in their own right and as a shortcut inside larger problems.

Property 1 — Rows and columns are interchangeable. The value is unchanged if the rows are written as columns and columns as rows (taking the transpose): $|A| = |A^{\mathsf{T}}|.$ Every property stated for rows therefore holds equally for columns.

Property 2 — Interchanging two rows (or two columns) reverses the sign. Swapping any two rows keeps the numerical value the same but flips its sign.

Property 3 — Two identical or proportional rows (or columns) make the determinant zero. If two rows are identical, or one is a constant multiple of another, the determinant is exactly $0$. This follows from Property 2: swapping two identical rows must both reverse the sign and leave the value unchanged, which is possible only if the value is $0$.

Property 4 — A common factor of a row (or column) comes outside. If every entry of one row (or column) has a common factor $k$, that $k$ can be taken out in front: $\begin{vmatrix} ka_1 & kb_1 & kc_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = k\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}.$

Property 5 — Adding a multiple of one row to another leaves the value unchanged. Replacing a row $R_i$ by $R_i + k R_j$ does not change the determinant. This is the tool used to create zeros in a row or column before expanding, and it is the single most useful simplification in the chapter. …