Square Matrix with Duplicate Rows has Zero Determinant

Theorem

If two rows (or columns) of a square matrix are the same, then its determinant is zero.

Corollary

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

Proof

From Determinant with Rows Transposed, if you swap over two rows of a matrix, the sign of its determinant changes.

If you swap over two identical rows of a matrix, then the sign of its determinant changes from $D$, say, to $-D$.

But the matrix is the same.

So $D = -D$ and so $D = 0$.

$\blacksquare$

Proof of Corollary

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

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 or column has the same determinant as that with two identical rows or columns.

That is, zero.

$\blacksquare$

Sources

• John F. Humphreys: A Course in Group Theory (1996): $\text{A}.2$: Theorem $\text{A}.10 \ (1)$