Square Matrix with Duplicate Columns has Zero Determinant/Corollary

Corollary to Square Matrix with Duplicate Columns has Zero Determinant
If a square matrix has a zero column, its determinant is zero.

Proof
If you add any column to a zero column, you get a square matrix with two identical columns.

From Multiple of Column Added to Column of Determinant, performing this operation does not change the value of the determinant.

So a square matrix with a zero column has the same determinant as that with two identical columns.

From Square Matrix with Duplicate Columns has Zero Determinant, that is zero.

Also see

 * Square Matrix with Duplicate Rows has Zero Determinant/Corollary