Square Matrix with Duplicate Columns has Zero Determinant

Theorem
If two columns of a square matrix over a commutative ring $\struct {R, +, \circ}$ are identical, then its determinant is zero.

Proof
Let $\mathbf A$ be a square matrix over $R$ with two identical columns.

Let $\mathbf A^\intercal$ denote the transpose of $\mathbf A$.

Then $\mathbf A^\intercal$ has two identical rows.

Then:

Also see

 * Square Matrix with Duplicate Rows has Zero Determinant