# 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.

### Corollary

If a square matrix has a zero column, 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:

 $\ds \map \det {\mathbf A}$ $=$ $\ds \map \det {\mathbf A^\intercal}$ Determinant of Transpose $\ds$ $=$ $\ds 0$ Square Matrix with Duplicate Rows has Zero Determinant

$\blacksquare$