Scaling a row scales the determinant

Statement

Let $K$ be a commutative ring, let $A$ be an $n \times n$ matrix over $K$, and let $c \in K$. Let $B$ be the matrix that results when one row of $A$ is multiplied by $c$. Then $\det B = c \det A$, where $\det$ is the determinant.

Remarks

This is a special case of linearity in each row. It describes the effect of the scaling row operation on the determinant. Scaling all $n$ rows gives $\det(cA) = c^n \det A$, see the determinant of a scalar multiple.

Explanations

L1 human

If we stretch one edge of a box by the factor $c$, then its volume is multiplied by $c$. The determinant behaves in the same way. For example, doubling one row doubles the determinant.