Adding a multiple of one row to another preserves 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 $i \ne j$, and let $B$ be the matrix that results when $c$ times row $j$ of $A$ is added to row $i$. Then $\det B = \det A$, where $\det$ is the determinant.

Remarks

By linearity in row $i$, $\det B = \det A + c \det A'$, where $A'$ has two equal rows; so $\det A' = 0$. This is the reason why Gaussian elimination computes determinants efficiently. The same holds for columns.

Explanations

L1 human

Adding a multiple of one row to another row shears the box spanned by the rows. A shear does not change the volume. So this row operation keeps the determinant.