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 $$D = 0$$.

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.