Determinant

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 $\mathbf{k}$.