Effect of Elementary Row Operations on Determinant

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

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

Take the elementary row operations:

Applying $\text {ERO} 1$ has the effect of multiplying $\map \det {\mathbf A}$ by $\lambda$.

Applying $\text {ERO} 2$ has no effect on $\map \det {\mathbf A}$.

Applying $\text {ERO} 3$ has the effect of multiplying $\map \det {\mathbf A}$ by $-1$.

Proof
From Elementary Row Operations as Matrix Multiplications, an elementary row operation on $\mathbf A$ is equivalent to matrix multiplication by the elementary row matrices corresponding to the elementary row operations.

From Determinant of Elementary Row Matrix, the determinants of those elementary row matrices are as follows:

Exchange Rows
Hence the result.