Exercise 629Topology › Topology of Mn(ℝ)L1Added 4 Nov 2026

Continuity of the determinant

#determinant#continuity#polynomial

Statement

Show that $\det:M_n(\mathbb{R})\to\mathbb{R}$ is continuous.

Hint

The determinant is a polynomial in the entries.

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Solution

Polynomial in entries. By the Leibniz formula,
$\det(A)=\sum_{\sigma\in S_n}\varepsilon(\sigma)\prod_{i=1}^n a_{i,\sigma(i)}.$
Each term is a product of continuous coordinate functions; the sum is continuous.

Conclusion. $\det$ is continuous, even polynomial in the entries.

Continuity of the determinant

Statement

Solution

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