\begin{definition}[Definition:Determinant/Matrix/Order 3]
Let $\mathbf A = \sqbrk a_3$ be a square matrix of order $3$.
That is, let:
:$\mathbf A = \begin {bmatrix}
a_{1 1} & a_{1 2} & a_{1 3} \\
a_{2 1} & a_{2 2} & a_{2 3} \\
a_{3 1} & a_{3 2} & a_{3 3}
\end {bmatrix}$
The determinant of $\mathbf A$ is given by:
:$\map \det {\mathbf A} = \begin {vmatrix}
a_{1 1} & a_{1 2} & a_{1 3} \\
a_{2 1} & a_{2 2} & a_{2 3} \\
a_{3 1} & a_{3 2} & a_{3 3}
\end {vmatrix}$
Then:
{{begin-eqn}}
{{eqn | l = \map \det {\mathbf A}
      | r = a_{1 1} \begin {vmatrix} a_{2 2} & a_{2 3} \\ a_{3 2} & a_{3 3} \end {vmatrix} - a_{1 2} \begin {vmatrix} a_{2 1} & a_{2 3} \\ a_{3 1} & a_{3 3} \end {vmatrix} + a_{1 3} \begin {vmatrix} a_{2 1} & a_{2 2} \\ a_{3 1} & a_{3 2} \end{vmatrix}
      | c = 
}}
{{eqn | r = \map \sgn {1, 2, 3} a_{1 1} a_{2 2}  a_{3 3}
      | c = 
}}
{{eqn | o = 
      | ro= +
      | r = \map \sgn {1, 3, 2} a_{1 1} a_{2 3}  a_{3 2}
      | c = 
}}
{{eqn | o = 
      | ro= +
      | r = \map \sgn {2, 1, 3} a_{1 2} a_{2 1}  a_{3 3}
      | c = 
}}
{{eqn | o = 
      | ro= +
      | r = \map \sgn {2, 3, 1} a_{1 2} a_{2 3}  a_{3 1}
      | c = 
}}
{{eqn | o = 
      | ro= +
      | r = \map \sgn {3, 1, 2} a_{1 3} a_{2 1}  a_{3 2}
      | c = 
}}
{{eqn | o = 
      | ro= +
      | r = \map \sgn {3, 2, 1} a_{1 3} a_{2 2}  a_{3 1}
      | c = 
}}
{{eqn | r = a_{1 1} a_{2 2}  a_{3 3}
      | c = 
}}
{{eqn | o = 
      | ro= -
      | r = a_{1 1} a_{2 3}  a_{3 2}
      | c = 
}}
{{eqn | o = 
      | ro= -
      | r = a_{1 2} a_{2 1}  a_{3 3}
      | c = 
}}
{{eqn | o = 
      | ro= +
      | r = a_{1 2} a_{2 3}  a_{3 1}
      | c = 
}}
{{eqn | o = 
      | ro= +
      | r = a_{1 3} a_{2 1}  a_{3 2}
      | c = 
}}
{{eqn | o = 
      | ro= -
      | r = a_{1 3} a_{2 2}  a_{3 1}
      | c = 
}}
{{end-eqn}}
and thence in a single expression as:
:$\ds \map \det {\mathbf A} = \frac 1 6 \sum_{i \mathop = 1}^3 \sum_{j \mathop = 1}^3 \sum_{k \mathop = 1}^3 \sum_{r \mathop = 1}^3 \sum_{s \mathop = 1}^3 \sum_{t \mathop = 1}^3 \map \sgn {i, j, k} \map \sgn {r, s, t} a_{i r} a_{j s} a_{k t}$
where $\map \sgn {i, j, k}$ is the sign of the permutation $\tuple {i, j, k}$ of the set $\set {1, 2, 3}$.
The values of the various instances of $\map \sgn {\lambda_1, \lambda_2, \lambda_3}$ are obtained by applications of Parity of K-Cycle.
\end{definition}