Expansion by Minors

To find the determinant of a $3 \times 3$ matrix, multiply each element in one row by its minor (the determinant of the matrix that remains when you cover the element's row and column), and combine using alternating signs. $$\begin{vmatrix} a 1 & b 1 & c 1 \\ a 2 & b 2 & c 2 \\ a 3 & b 3 & c 3 \end{vmatrix} = a 1 \begin{vmatrix} b 2 & c 2 \\ b 3 & c 3 \end{vmatrix} b 1 \begin{vmatrix} a 2 & c 2 \\ a 3 & c 3 \end{vmatrix} + c 1 \begin{vmatrix} a 2 & b 2 \\ a 3 & b 3 \end{vmatrix}$$.

Evaluate $\begin{vmatrix} 2 & 1 & 0 \\ 3 & 1 & 4 \\ 2 & 5 & 6 \end{vmatrix} = 2\begin{vmatrix} 1 & 4 \\ 5 & 6 \end{vmatrix} 1\begin{vmatrix} 3 & 4 \\ 2 & 6 \end{vmatrix} + 0\begin{vmatrix} 3 & 1 \\ 2 & 5 \end{vmatrix} = 2( 26) 1(26) + 0 = 78$. Evaluate $\begin{vmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \\ 0 & 1 & 2 \end{vmatrix} = 1\begin{vmatrix} 4 & 6 \\ 1 & 2 \end{vmatrix} 3\begin{vmatrix} 2 & 6 \\ 0 & 2 \end{vmatrix} + 5\begin{vmatrix} 2 & 4 \\ 0 & 1 \end{vmatrix} = 1(2) 3(4) + 5(2) = 0$.

To crack a 3x3 matrix, break it into smaller 2x2 puzzles. Choose a row, typically the first one. For each element, cover its row and column to find its 'minor'—the determinant of the 2x2 grid that remains. Multiply the element by its minor, then combine the results using an alternating plus minus plus pattern. It's that simple!