Effect of Elementary Column 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 column operations:

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

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

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

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

From Determinant of Elementary Column Matrix, the determinants of those elementary column matrices are as follows:

Exchange Columns
Hence the result.

Also see

 * Effect of Elementary Row Operations on Determinant