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.

$\blacksquare$

Also see

• Square Matrix with Duplicate Rows has Zero Determinant: Corollary

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): $\text{A}.2$: Linear algebra and determinants: Theorem $\text{A}.10 \ (1)$
• 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.6$ Determinant and trace: Fact $1.7 \ \text {(e)}$