Properties of Determinants

Evaluating every determinant by direct expansion is reliable but slow. A set of standard 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 used constantly in the CHSE Odisha commerce mathematics paper, both as a topic in their own right and as a shortcut inside larger problems.

Property 1 — Interchange of rows and columns. The value of a determinant is unchanged if its rows are written as columns and its columns as rows (this operation is called taking the transpose). $|A| = |A^{T}|.$

Property 2 — Interchanging two rows (or two columns) reverses the sign. If any two rows of a determinant are interchanged, the numerical value stays the same but its sign flips.

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

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

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