The determinant is a scalar value that can be computed from a square matrix. It provides important information about the matrix, such as whether it is invertible and its effect on volume when used in transformations.
\[ |A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = (ad - bc) \]
This is the determinant of a 2×2 matrix. Multiply diagonally and subtract: \( a \cdot d - b \cdot c \).
\[ |A| = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a \cdot \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \cdot \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \cdot \begin{vmatrix} d & e \\ g & h \end{vmatrix} \]
This is the determinant of a 3×3 matrix using cofactor expansion along the first row.
\[ = a(ei - fh) - b(di - fg) + c(dh - eg) \]
Each 2×2 sub-determinant is expanded here using the formula for a 2×2 determinant.
\[ = aei + bfg + c \cdot dh - ceg - b \cdot di - afh \]
Final expanded form combining all terms.