TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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The determinant of a triangular matrix is the product of its diagonal entries

Statement

If $A=(a_{ij})\in M_n(R)$ is upper or lower triangular over a commutative ring, with $n\ge1$ , then $\det(A)=\prod_{i<n}a_{ii}.$

Facts & Assumptions

Given: A triangular $n\times n$ matrix $A$ over a commutative ring.

[L1]

Upper triangular means $a_{ij}=0$ for $i>j$ , while lower triangular means $a_{ij}=0$ for $i<j$ (Upper triangular, lower triangular and diagonal square matrices over a commutative ring).

[L2]

Determinant is the sum of the signed products $\prod_i a_{\sigma(i),i}$ (For $n\ge1$ , the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix).

[L3]

The identity permutation has sign $1$ (Inversions, inversion number, the sign $\operatorname{sgn}(\sigma)=(-1)^{\operatorname{inv}(\sigma)}$ , and even and odd permutations).

Proof

technique · direct

1.1

For an upper triangular matrix, a nonzero Leibniz term requires $\sigma(i)\le i$ for every $i$ . A permutation with these inequalities is the identity: starting at $i=0$ and proceeding upward, bijectivity forces $\sigma(i)=i$ . Thus every nonidentity term contains a zero entry.

L1L2L3

2.1

The identity term has sign $1$ and equals $\prod_{i<n}a_{ii}$ , so it is the whole determinant.

step 1.1L2L3

3.1

For a lower triangular matrix the same argument uses $\sigma(i)\ge i$ and proceeds downward, again forcing the identity. The argument includes $n=1$ and makes no cancellation assumption, so zero divisors and the zero ring cause no exception.

step 2.1L1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 66 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

S. New, MATH 146 Linear Algebra 1 Lecture Notes, Ch. 4 (standard reference, not scraped)
D. Margalit and J. Rabinoff, Interactive Linear Algebra, §4.1 (standard reference, not scraped)