Square Matrix with Duplicate Rows has Zero Determinant/Corollary

Corollary to Square Matrix with Duplicate Rows has Zero Determinant

If a square matrix has a zero row, its determinant is zero.

Proof

If you add any row to a zero row, you get a square matrix with two identical rows.

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

So a square matrix with a zero row has the same determinant as that with two identical rows.

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

$\blacksquare$

Also see

• Square Matrix with Duplicate Columns has Zero Determinant: Corollary

Sources

• 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 $1.7 \ \text {(e)}$