Let $A \in \mathbb{R}^{N \times N}$ be a
D6160: Real square matrix such that
| (i) |
$N \in \{ 2, 3, 4, \ldots \}$ is a D5094: Positive integer
|
| (ii) |
$a_1, \ldots, a_N \in \mathbb{R}^{N \times 1}$ are each a D5200: Real column matrix
|
| (iii) |
$b_1, \ldots, b_N \in \mathbb{R}^{1 \times N}$ are each a D5201: Real row matrix
|
| (iv) |
\begin{equation}
A
=
\begin{bmatrix}
a_1 & a_2 & \cdots & a_N
\end{bmatrix}
\end{equation}
|
| (v) |
\begin{equation}
A
=
\begin{bmatrix}
b_1 \\
b_2 \\
\vdots \\
b_N
\end{bmatrix}
\end{equation}
|
| (vi) |
$\pi : \{ 1, \ldots, N \} \to \{ 1, \ldots, N \}$ is a D6177: Standard transposition
|
| (vii) |
\begin{equation}
X
=
\begin{bmatrix}
a_{\pi(1)} & a_{\pi(2)} & \cdots & a_{\pi(N)}
\end{bmatrix}
\end{equation}
|
| (viii) |
\begin{equation}
Y
=
\begin{bmatrix}
b_{\pi(1)} \\
b_{\pi(2)} \\
\vdots \\
b_{\pi(N)}
\end{bmatrix}
\end{equation}
|
Then
| (1) |
\begin{equation}
\text{Det} X
= - \text{Det} A
\end{equation}
|
| (2) |
\begin{equation}
\text{Det} Y
= - \text{Det} A
\end{equation}
|
