Determinant with Row Multiplied by Constant/Proof 1

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Theorem

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.

Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.


Let $\mathbf B$ be the matrix resulting from one row of $\mathbf A$ having been multiplied by a constant $c$.


Then:

$\map \det {\mathbf B} = c \map \det {\mathbf A}$


That is, multiplying one row of a square matrix by a constant multiplies its determinant by that constant.


Proof

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.

Let $e$ be the elementary row operation that multiplies rows $i$ by the scalar$c$.

Let $\mathbf B = \map e {\mathbf A}$.

Let $\mathbf E$ be the elementary row matrix corresponding to $e$.

From Elementary Row Operations as Matrix Multiplications:

$\mathbf B = \mathbf E \mathbf A$

From Determinant of Elementary Row Matrix: Exchange Rows:

$\map \det {\mathbf E} = c$

Then:

\(\ds \map \det {\mathbf B}\) \(=\) \(\ds \map \det {\mathbf E \mathbf A}\) Determinant of Matrix Product
\(\ds \) \(=\) \(\ds c \map \det {\mathbf A}\) as $\map \det {\mathbf E} = c$

Hence the result.

$\blacksquare$


Sources

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