Multiple of Row Added to Row of Determinant

Theorem
Let $\mathbf A = \begin {bmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ \vdots & \vdots & \ddots &  \vdots \\ a_{r 1} & a_{r 2} & \cdots & a_{r n} \\ \vdots & \vdots & \ddots &  \vdots \\ a_{s 1} & a_{s 2} & \cdots & a_{s n} \\ \vdots & \vdots & \ddots &  \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \\ \end {bmatrix}$ be a square matrix of order $n$.

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

Let $\mathbf B = \begin{bmatrix} a_{1 1}            & a_{1 2}             & \cdots &             a_{1 n} \\ \vdots            & \vdots              & \ddots &              \vdots \\ a_{r 1} + k a_{s 1} & a_{r 2} + k a_{s 2} & \cdots & a_{r n} + k a_{s n} \\ \vdots            & \vdots              & \ddots &              \vdots \\ a_{s 1}            & a_{s 2}             & \cdots &             a_{s n} \\ \vdots            & \vdots              & \ddots &              \vdots \\ a_{n 1}            & a_{n 2}             & \cdots &             a_{n n} \\ \end{bmatrix}$.

Then $\map \det {\mathbf B} = \map \det {\mathbf A}$.

That is, the value of a determinant remains unchanged if a constant multiple of any row is added to any other row.

Also see

 * Multiple of Column Added to Column of Determinant