The determinant of a matrix is a number which tells us something about the properties of the matrix. For example, if the matrix represents a 2-dimensional linear transformation, then the determinant will tell us the ratio by which areas are scaled, while for a 3-dimensional linear transformation, it will give the volume ratio.
Often the determinant of \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \] is written as \[ \begin{vmatrix} a & b \\ c & d \end{vmatrix} \quad\text{or}\quad \det \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]
For this \(2 \times 2\)-matrix, the determinant is \(\det A = ad - bc\). In this case the determinant is the area of a parallelogram with sides given by the vectors \[ \begin{pmatrix} a \\ c \end{pmatrix} \quad\text{and}\quad \begin{pmatrix} b \\ d \end{pmatrix}, \] which is the image of the unit square with sides \(\mathbf{i}=\bigl( \begin{smallmatrix}1\\0\end{smallmatrix}\bigr)\) and \(\mathbf{j}=\bigl( \begin{smallmatrix}0\\1\end{smallmatrix}\bigr)\).
For a \(3 \times 3\)-matrix, we can find the determinant as follows: \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \begin{vmatrix} d & e \\ g & h \end{vmatrix}. \] (This is called the Laplace expansion or the cofactor expansion of the determinant. This generalises to \(n \times n\)-matrices.) In this case, the determinant is the volume of the parallelepiped with edges given by the vectors \[ \begin{pmatrix} a \\ d \\ g \end{pmatrix}, \begin{pmatrix} b \\ e \\ h \end{pmatrix}, \quad\text{and}\quad \begin{pmatrix} c \\ f \\ i \end{pmatrix}, \] which is the image of the unit cube with sides \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\).
