The Determinant

2 by 2 Matrix The determinant of a matrix is its value, it has many applications in knot theory as well as algebra, as in Cramer's Rule. Cramer's Rule allows you to solve n equations in n variables.
The determinant of the matrix at right can be determined with the expression a1b2 - a2b1
3 by 3 Matrix3 by 3 Matrix with row and column crossed outWhen you try to solve a 3 x 3 matrix however, things get more complicated. Each element of the matrix at left is associated with a 2 x 2 matrix. The method for finding that 2 x 2 matrix is shown at right for the element a2. When this is done you get the remaining 2 x 2 matrix.2 by 2 Matrix in 3 by 3 Matrix

Alternating + and - matrix pattern Now, in order to find the determinant of a 3 x 3 matrix it is nescessary to find the determinants of the 2 x 2 matricies for all the elements in any column, multiply them with their respective elements and either add or subtract them according to the diagram at left.