3.4. Determinants

Given an \(N\times N\) matrix of elements (numbers, functions), a determinant is a specific linear combination of \(N!\) products of the elements, with +1 and -1 as the linear combination coefficients. For a matrix \(A\), the determinant is denoted \(\mathrm{det}A\) or \(|A|\).

The determinant of a 2x2 matrix is

(3.52)\[\begin{split}\mathrm{det} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{vmatrix} a & b\\ c & d \end{vmatrix} = ad-bc\end{split}\]

For a 3x3 matrix, the determinant can be written as a linear combination of determinants of \(2\times2\) sub-matrices:

(3.53)\[\begin{split}\mathrm{det} \begin{pmatrix} a & b & c\\ d & e & f\\ g & h & i \end{pmatrix} = a \begin{vmatrix} e & f\\ h & i \end{vmatrix} - b \begin{vmatrix} d & f\\ g & i \end{vmatrix} + c \begin{vmatrix} d & e\\ g & h \end{vmatrix}\end{split}\]

Multiplied out, this gives

(3.54)\[= aei-afh-bdi+bfg+cdh-ceg\]

A \(4\times4\) determinant has 24 products.

Some important properties of determinants are:

• If two rows or two columns are equal, the determinant is zero.

• Swapping two columns, or swapping two rows, inverts the sign of the determinant.

• If the matrix is transposed, i.e. if the columns are converted to rows and vice versa, the value of the determinant remains unchanged.