\begin{definition}[Definition:Determinant/Matrix/Definition 2]
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
That is, let:
:$\mathbf A = \begin {bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\
 \vdots &  \vdots & \ddots &  \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n} \\
\end {bmatrix}$
The '''determinant''' of $\mathbf A$ is defined as follows:
For $n = 1$, the order $1$ determinant is defined as:
{{:Definition:Determinant of Order 1}}
For $n > 1$, the determinant of order $n$ is defined recursively as:
:$\ds \map \det {\mathbf A} := \begin {vmatrix}
a_{1 1} & a_{1 2} & a_{1 3} & \cdots & a_{1 n} \\
a_{2 1} & a_{2 2} & a_{2 3} & \cdots & a_{2 n} \\
a_{3 1} & a_{3 2} & a_{3 3} & \cdots & a_{3 n} \\
 \vdots &  \vdots &  \vdots & \ddots &  \vdots \\
a_{n 1} & a_{n 2} & a_{n 3} & \cdots & a_{n n} \\
\end {vmatrix} = a_{1 1} \begin {vmatrix}
a_{2 2} & a_{2 3} & \cdots & a_{2 n} \\
a_{3 2} & a_{3 3} & \cdots & a_{3 n} \\
 \vdots &  \vdots & \ddots &  \vdots \\
a_{n 2} & a_{n 3} & \cdots & a_{n n} \\
\end {vmatrix} - a_{1 2} \begin {vmatrix}
a_{2 1} & a_{2 3} & \cdots & a_{2 n} \\
a_{3 1} & a_{3 3} & \cdots & a_{3 n} \\
 \vdots &  \vdots & \ddots &  \vdots \\
a_{n 1} & a_{n 3} & \cdots & a_{n n} \\
\end {vmatrix} + \cdots + \paren {-1}^{n + 1} a_{1 n} \begin {vmatrix}
a_{2 1} & a_{2 2} & \cdots & a_{2, n - 1} \\
a_{3 1} & a_{3 3} & \cdots & a_{3, n - 1} \\
 \vdots &  \vdots & \ddots &       \vdots \\
a_{n 1} & a_{n 3} & \cdots & a_{n, n - 1} \\
\end {vmatrix}$
\end{definition}