Determinants: (lesson 1 of 2)

## Introduction to Determinant

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|
$