This page supports the main stream of the web site by providing/reminding some information regading determinants and inverse matrices. We demonstrate capabilities of MuPad for this topic.
General definition of determinant is difficult and counterintuitive. We define it recursively. For a \( 1 \times 1 \) matrix that consists of one element, \( {\bf A} = [a] , \) its determinant is \( \det {\bf A} = \left\vert {\bf A} \right\vert = a . \) Note that it is a custom to use two notations for determinants: one is to write ``det'' in front of a square matrix, and another one is to embrase the square array with vertical lines. For a \( 2\times 2 \) matrix \( {\bf A} = \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right] , \) its determinant is \( \det {\bf A} = \left\vert {\bf A} \right\vert = a_{11} a_{22} - a_{12} a_{21} . \)
In general, we define the determinant (the term determinant was first introduced by the German mathematician Carl Friedrich Gauss in 1801) using cofactor expansion. Therefore, we need a definition.
If A is a square matrix, then the minor (the term was introduced by James Sylvester in 1850) of entry \( a_{ij} \) is denoted by \( M_{ij} \) and is defined to be the determinant of the submatrix that remains after the i-th row and j-th column are deleted from A. The number \( (-1)^{i+j} M_{ij} \) is denoted by \( C_{ij} \) and is called the cofactor of entry \( a_{ij} . \) The matrix of cofactors of a square matrix A is called the adjugate of A.
There is a standard comamnd in MuPad to evaluate the determinant of a square matrix: