Let $A \in \mathbb{C}^{N \times N}$ be a
D6159: Complex square matrix such that
| (i) |
$N \in \{ 1, 2, 3, \ldots \}$ is a D5094: Positive integer
|
| (ii) |
$a_1, \ldots, a_N \in \mathbb{C}^{N \times 1}$ are each a D5689: Complex column matrix
|
| (iii) |
$b_1, \ldots, b_N \in \mathbb{C}^{1 \times N}$ are each a D5688: Complex row matrix
|
| (iv) |
\begin{equation}
A
=
\begin{bmatrix}
a_1 & \cdots & a_k & \cdots & a_N
\end{bmatrix}
\end{equation}
|
| (v) |
\begin{equation}
A
=
\begin{bmatrix}
b_1 \\
\vdots \\
b_k \\
\vdots \\
b_N
\end{bmatrix}
\end{equation}
|
| (vi) |
$\lambda \in \mathbb{C}$ is a D1207: Complex number
|
| (vii) |
\begin{equation}
X
=
\begin{bmatrix}
a_1 & \cdots & \lambda a_k & \cdots & a_N
\end{bmatrix}
\end{equation}
|
| (viii) |
\begin{equation}
Y
=
\begin{bmatrix}
b_1 \\
\vdots \\
\lambda b_k \\
\vdots \\
b_N
\end{bmatrix}
\end{equation}
|
Using
R5517: Cofactor partition for a complex square matrix, we have
\begin{equation}
\text{Det} Y
= \sum_{n = 1}^N Y_{k, n} C_{k, n}
= \sum_{n = 1}^N \lambda A_{k, n} C_{k, n}
= \lambda \sum_{n = 1}^N A_{k, n} C_{k, n}
= \lambda \text{Det} A
\end{equation}
The case for $X$ (where a column is scaled) is proven anologously using the same result. $\square$
