Square Matrix with Duplicate Columns has Zero Determinant

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Theorem

If two columns of a square matrix over a commutative ring $\struct {R, +, \circ}$ are identical, then its determinant is zero.


Corollary

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


Proof

Let $\mathbf A$ be a square matrix over $R$ with two identical columns.

Let $\mathbf A^\intercal$ denote the transpose of $\mathbf A$.

Then $\mathbf A^\intercal$ has two identical rows.

Then:

\(\ds \map \det {\mathbf A}\) \(=\) \(\ds \map \det {\mathbf A^\intercal}\) Determinant of Transpose
\(\ds \) \(=\) \(\ds 0\) Square Matrix with Duplicate Rows has Zero Determinant

$\blacksquare$


Also see


Sources

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