In the following we assume we have a square matrix $(m=n)$. The determinant of a matrix $A$ will be denoted by $\det(A)$ or $|A|$.

Determinant of a 2 $\times$ 2 matrix

Assuming $A$ is an arbitrary 2 $\times$ 2 matrix $A$, where the elements are given by: $ A = \left( {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}\\ {{a_{21}}}&{{a_{22}}} \end{array}} \right) $

then the determinant of this matrix is as follows:

$ \det (A) = \left| A \right| = \left| {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}\\ {{a_{21}}}&{{a_{22}}} \end{array}} \right| = {a_{11}}{a_{22}} - {a_{21}}{a_{12}} $

Determinant of a 3 $\times$ 3 matrix

The determinant of a 3 $\times$ 3 matrix is a little more tricky and is found as follows ( for this case assume $A$ is an arbitrary 3 $\times$ 3 matrix $A$, where the elements are given below)

$ A = \left( {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right) $

then the determinant of this matrix is as follows:

$ \det (A) = \left| {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right| = {a_{11}}\left| {\begin{array}{*{20}{c}} {{a_{22}}}&{{a_{23}}}\\ {{a_{32}}}&{{a_{33}}} \end{array}} \right| - {a_{12}}\left| {\begin{array}{*{20}{c}} {{a_{21}}}&{{a_{23}}}\\ {{a_{31}}}&{{a_{33}}} \end{array}} \right| + {a_{13}}\left| {\begin{array}{*{20}{c}} {{a_{21}}}&{{a_{22}}}\\ {{a_{31}}}&{{a_{32}}} \end{array}} \right| $