Properties of Determinants

Expanding a $3\times3$ determinant term by term every time is slow and error-prone. A handful of properties, applied as row/column operations, let us simplify a determinant — or spot that it is zero — often without expanding it fully at all.

1. Interchange property. If any two rows (or two columns) of a determinant are interchanged, the value of the determinant changes sign but keeps the same magnitude.
2. Equal rows/columns property. If two rows (or two columns) of a determinant are identical, the determinant equals $0$. (This follows directly from property 1: interchanging the two identical rows must both change the sign and leave the determinant unchanged, which is only possible if the value is $0$.)
3. Common factor property. If every element of one row (or column) has a common factor $k$, that factor can be taken outside the determinant: $\begin{vmatrix}ka&kb\\c&d\end{vmatrix}=k\begin{vmatrix}a&b\\c&d\end{vmatrix}$
4. Row/column operation property. The value of a determinant is unchanged if a multiple of one row (or column) is added to another row (or column), e.g. $R_1 \to R_1 + kR_2$. This is the single most useful property in practice — it is used to create zeros that make expansion faster, and to reveal hidden linear relationships between rows.
5. Sum property. If every element of one row (or column) is written as a sum of two terms, the determinant splits into the sum of two determinants, each keeping the other rows/columns unchanged. …