# Definition:Determinant/Matrix/Examples

## Examples of Determinant

### Determinant of Order 1

This is the trivial case:

$\begin{vmatrix} a_{11} \end{vmatrix} = \operatorname{sgn} \left({1}\right) a_{1 1} = a_{1 1}$

Thus the determinant of a single number is that number itself.

### Determinant of Order 2

$\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = \map \sgn {1, 2} a_{1 1} a_{2 2} + \map \sgn {2, 1} a_{1 2} a_{2 1} = a_{1 1} a_{2 2} - a_{1 2} a_{2 1}$

### Determinant of Order 3

Let:

$\det \left({\mathbf A}\right) = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$

Then:

 $\displaystyle \det \left({\mathbf A}\right) = \ \$ $\displaystyle \operatorname{sgn} \left({1, 2, 3}\right) a_{1 1} a_{2 2} a_{3 3}$ $+$ $\displaystyle \operatorname{sgn} \left({1, 3, 2}\right) a_{1 1} a_{2 3} a_{3 2}$ $\displaystyle$ $+$ $\displaystyle \operatorname{sgn} \left({2, 1, 3}\right) a_{1 2} a_{2 1} a_{3 3}$ $\displaystyle$ $+$ $\displaystyle \operatorname{sgn} \left({2, 3, 1}\right) a_{1 2} a_{2 3} a_{3 1}$ $\displaystyle$ $+$ $\displaystyle \operatorname{sgn} \left({3, 1, 2}\right) a_{1 3} a_{2 1} a_{3 2}$ $\displaystyle$ $+$ $\displaystyle \operatorname{sgn} \left({3, 2, 1}\right) a_{1 3} a_{2 2} a_{3 1}$ $\displaystyle = \ \$ $\displaystyle a_{1 1} a_{2 2} a_{3 3}$ $-$ $\displaystyle a_{1 1} a_{2 3} a_{3 2}$ $\displaystyle$ $-$ $\displaystyle a_{1 2} a_{2 1} a_{3 3}$ $\displaystyle$ $+$ $\displaystyle a_{1 2} a_{2 3} a_{3 1}$ $\displaystyle$ $+$ $\displaystyle a_{1 3} a_{2 1} a_{3 2}$ $\displaystyle$ $-$ $\displaystyle a_{1 3} a_{2 2} a_{3 1}$

and thence in a single expression as:

$\displaystyle \det \left({\mathbf A}\right) = \frac 1 6 \sum_{i \mathop = 1}^3 \sum_{j \mathop = 1}^3 \sum_{k \mathop = 1}^3 \sum_{r \mathop = 1}^3 \sum_{s \mathop = 1}^3 \sum_{t \mathop = 1}^3 \operatorname{sgn} \left({i, j, k}\right) \operatorname{sgn} \left({r, s, t}\right) a_{i r} a_{j s} a_{k t}$

where $\operatorname{sgn} \left({i, j, k}\right)$ is the sign of the permutation $\left({i, j, k}\right)$ of the set $\left\{{1, 2, 3}\right\}$.

The values of the various instances of $\operatorname{sgn} \left({\lambda_1, \lambda_2, \lambda_3}\right)$ are obtained by applications of Parity of K-Cycle.