# Determinants

Every square matrix, ** A**, is assigned a particular scalar quantity called the **determinant of A**, denoted by **|A|**or by **det (A)**.

|A| =

## Determinant of Order One

|a_{11}| = a_{11}

|5| = 5

## Determinant of Order Two

= **a _{11} a _{22} − a _{12} a _{21} **

## Determinant of Order Three

Consider an arbitrary 3 x 3 matrix, A = (a_{ij}). The determinant of A is defined as follows:

=

**a _{11}
a_{22}
a_{33} +
a_{12}
a_{23 }
a _{31} +
a_{13}
a_{21 }
a_{32} - **

**- a _{13}
a_{22}
a_{31} -
a_{12}
a_{21}
a_{ 33 } -
a_{11}
a_{23}
a_{32.}**

=

**3 · 2 · 4 + 2 · (-5) · (-2) + 1 · 0 · 1 - **

**- 1 · 2 · (-2) - 2 · 0 · 4 - 3 · (-5) · 1 =**

**= 24 + 20 + 0 - (-4) - 0 - (-15) =**

**= 44 + 4 + 15 = 63 **

Note that there are six products, each consisting of three elements in the matrix. Three of the products appear with a positive sign (they preserve their sign) and three with a negative sign (they change their sign).