Determinant of Rescaling Matrix/Corollary
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Corollary to Determinant of Rescaling Matrix
Let $\mathbf A$ be a square matrix of order $n$.
Let $\lambda$ be a scalar.
Let $\lambda \mathbf A$ denote the scalar product of $\mathbf A$ by $\lambda$.
Then:
- $\map \det {\lambda \mathbf A} = \lambda^n \map \det {\mathbf A}$
where $\det$ denotes determinant.
Proof
For $1 \le k \le n$, let $e_k$ be the elementary row operation that multiplies row $k$ of $\mathbf A$ by $\lambda$.
By definition of the scalar product, $\lambda \mathbf A$ is obtained by multiplying every row of $\mathbf A$ by $\lambda$.
That is the same as applying $e_k$ to $\mathbf A$ for each of $k \in \set {1, 2, \ldots, n}$.
Let $\mathbf E_k$ denote the elementary row matrix corresponding to $e_k$.
By Determinant of Elementary Row Matrix: Scale Row:
- $\map \det {\mathbf E_k} = \lambda$
Then we have:
\(\ds \lambda \mathbf A\) | \(=\) | \(\ds \prod_{k \mathop = 1}^n \mathbf E_k \mathbf A\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \det {\lambda \mathbf A}\) | \(=\) | \(\ds \map \det {\prod_{k \mathop = 1}^n \mathbf E_k \mathbf A}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\prod_{k \mathop = 1}^n \map \det {\mathbf E_k} } \map \det {\mathbf A}\) | Determinant of Matrix Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\prod_{k \mathop = 1}^n \lambda} \map \det {\mathbf A}\) | Determinant of Elementary Row Matrix: Scale Row | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda^n \map \det {\mathbf A}\) |
$\blacksquare$
Sources
- 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.6$ Determinant and trace: Proposition $1.10$