Determinants of Order Three (3×3 Determinants)

A determinant of order 3 is formed from a $3\times 3$ square matrix and is evaluated by expansion along any row or any column — every row and every column gives the same final value, which is one of the most useful facts about determinants (it lets a student always pick the row or column with the most zeros, to reduce the arithmetic).

For $A=\begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix},$ expansion along the first row gives $|A| = a_1\begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix} - b_1\begin{vmatrix} a_2 & c_2 \\ a_3 & c_3 \end{vmatrix} + c_1\begin{vmatrix} a_2 & b_2 \\ a_3 & b_3 \end{vmatrix}.$

Each term is built the same way: take an entry from the chosen row, delete the row and column that entry sits in (leaving a smaller $2\times 2$ determinant, called that entry's minor), multiply the entry by its minor, and attach a $+$ or $-$ sign that alternates across the row, starting with $+$ for the first entry.

The sign pattern for a $3\times 3$ determinant is fixed and worth memorising: $\begin{vmatrix} + & - & + \\ - & + & - \\ + & - & + \end{vmatrix}.$ Expansion along the second row, for instance, therefore carries a minus sign on its first term: $|A| = -a_2\begin{vmatrix} b_1 & c_1 \\ b_3 & c_3 \end{vmatrix} + b_2\begin{vmatrix} a_1 & c_1 \\ a_3 & c_3 \end{vmatrix} - c_2\begin{vmatrix} a_1 & b_1 \\ a_3 & b_3 \end{vmatrix}.$ …