# Square Matrix with Duplicate Columns has Zero Determinant/Corollary

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## 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

## 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 $1.7 \ \text {(e)}$