Determinant of a transpose

Statement

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$. Then $\det A^T = \det A$, where $A^T$ is the transpose of $A$ and $\det$ is the determinant.

Remarks

In the Leibniz formula, the term of $\sigma$ for $A^T$ equals the term of $\sigma^{-1}$ for $A$, and $\sgn(\sigma^{-1}) = \sgn(\sigma)$. So each property of the determinant for rows also holds for columns.

Explanations

L1 human

Exchanging rows and columns does not change the determinant. So we can expand, reduce or test a determinant by columns as well as by rows. Every rule for rows has a twin rule for columns.