Determinant of a triangular matrix

Statement

Let $K$ be a commutative ring and let $A = (a_{ij})$ be an $n \times n$ upper or lower triangular matrix over $K$. Then $$\det A = a_{11} a_{22} \cdots a_{nn},$$ where $\det$ is the determinant.

Remarks

In the Leibniz formula, all terms except the term of the identity permutation contain an entry below, or above, the diagonal, so they are zero. Row reduction to a triangular matrix, with this rule, is a practical way to compute determinants. A triangular matrix over a field is invertible if and only if all its diagonal entries are nonzero.

Explanations

L1 human

For a triangular matrix, the determinant is the product of the diagonal entries. The entries above the diagonal do not matter. For example, $\det \begin{pmatrix} 2 & 7 \\ 0 & 3 \end{pmatrix} = 6$.