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Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof of the determinant of a triangular matrix by the Leibniz formula

Builds on Leibniz formula for determinants Upper triangular matrix Lower triangular matrix Sign of a permutation

Proof

By the Leibniz formula, $\det A = \sum_{\sigma \in S_n} \sgn(\sigma) \prod_{i=1}^{n} a_{i, \sigma(i)}$ .
Let $\sigma \in S_n$ with $\sigma(i) \ge i$ for all $i$ . Since $\sigma$ is a bijection, $\sum_{i=1}^{n} \sigma(i) = \sum_{i=1}^{n} i$ .
So the nonnegative numbers $\sigma(i) - i$ have sum $0$ . Hence $\sigma(i) = i$ for all $i$ , and $\sigma$ is the identity.
In the same way, if $\sigma(i) \le i$ for all $i$ , then $\sigma$ is the identity.
Let $A$ be upper triangular, so $a_{ij} = 0$ for $i > j$ . Let $\sigma$ be not the identity.
By step 2 and step 3, there is an $i$ with $\sigma(i) < i$ . Then $a_{i, \sigma(i)} = 0$ , so the term of $\sigma$ in step 1 is $0$ .
Let $A$ be lower triangular, so $a_{ij} = 0$ for $i < j$ . Let $\sigma$ be not the identity.
By step 4, there is an $i$ with $\sigma(i) > i$ . Then $a_{i, \sigma(i)} = 0$ , so the term of $\sigma$ in step 1 is $0$ .
In both cases only the identity contributes to the sum. Its sign is $1$ . For $n \ge 2$ , this follows from the definition of the sign, with zero transpositions. For $n = 1$ , it is the convention of the Leibniz formula.
So $\det A = a_{11} a_{22} \cdots a_{nn}$ .

Idea

Each term of the Leibniz formula picks one entry from each row and each column. Any permutation other than the identity must step below the diagonal somewhere and also above it somewhere. In a triangular matrix one of those entries is zero, so only the diagonal product survives.

Reading path

To understand this page from the ground up, read these 101 pages in order. The foundations come first. The path takes the proofs Proof of the decomposition of a permutation into transpositions, Proof that the symmetric group is generated by transpositions, Proof of the commutativity of addition by induction, Proof that equivalence classes form a partition, Proof that permuting rows multiplies the determinant by the sign, Proof that disjoint cycles commute, Proof that a matrix with two equal rows has determinant zero, Proof of strong induction from ordinary induction, Proof of the laws of exponents in a group, Proof that an intersection of subgroups is a subgroup, Proof that the determinant of the identity matrix is one, Proof of the decomposition of a permutation into disjoint cycles, Proof of the well-ordering principle from strong induction, Proof of the division algorithm, Proof of the Leibniz formula from the defining properties of the determinant, Proof of the uniqueness of inverses, Proof that the determinant is multilinear in the rows, Proof of the subgroup criterion.

Binary operation definition
Cycle notation concept
Definition by recursion concept
Divisibility definition
Element definition
Group concept
Identity matrix object
Mathematical induction concept
Matrix definition
Multilinear map concept
Natural numbers object
Peano axioms concept
Permutation concept
Proposition concept
Transposition concept
Vector space definition
Identity element definition
Set definition
Subgroup definition
Symmetric group definition
Square matrix definition
Successor function definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Linear map definition
Inverse element definition
Empty set object
Family of sets definition
Ordered pair definition
Transposition definition
Proof that an intersection of subgroups is a subgroup proof
Intersection definition
Lower triangular matrix definition
Union definition
Upper triangular matrix definition
Set-builder notation concept
Addition of natural numbers definition
Biconditional definition
The successor function is injective statement
Zero is not a successor statement
Existential quantifier definition
Universal quantifier definition
Multilinear map definition
Proof of the uniqueness of inverses proof
Cartesian product definition
Disjoint sets definition
Subset definition
Proof of the commutativity of addition by induction proof
Axiom of extensionality statement
Union of a family of sets definition
Order on the natural numbers definition
Uniqueness quantifier definition
Alternating multilinear map definition
Uniqueness of inverses statement
Binary relation definition
Partition of a set definition
Determinant definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Proof of the subgroup criterion proof
Proof of the laws of exponents in a group proof
Proof that a matrix with two equal rows has determinant zero proof
Function definition
Proof that the determinant of the identity matrix is one proof
Proof that the determinant is multilinear in the rows proof
Strong induction statement
Equivalence relation definition
Subgroup criterion statement
Determinant of a matrix with two equal rows statement
Laws of exponents in a group statement
Determinant of the identity matrix statement
Determinant is multilinear in the rows statement
Injective function definition
Proof of the well-ordering principle from strong induction proof
Intersection of subgroups is a subgroup statement
Proof that disjoint cycles commute proof
Equivalence class definition
Well-ordering principle statement
Subgroup generated by a subset definition
Disjoint cycles commute statement
Proof that equivalence classes form a partition proof
Proof of the division algorithm proof
Equivalence classes form a partition statement
Division algorithm statement
Proof of the decomposition of a permutation into disjoint cycles proof
Decomposition of a permutation into disjoint cycles statement
Proof of the decomposition of a permutation into transpositions proof
Decomposition of a permutation into transpositions statement
Sign of a permutation definition
Proof that the symmetric group is generated by transpositions proof
The symmetric group is generated by transpositions statement
Proof that permuting rows multiplies the determinant by the sign proof
Permuting rows multiplies the determinant by the sign statement
Proof of the Leibniz formula from the defining properties of the determinant proof
Leibniz formula for determinants statement
Proof of the determinant of a triangular matrix by the Leibniz formula proof

Where this leads

43 pages build on this one.

Further reading

Web https://en.wikipedia.org/wiki/Triangular_matrix, Properties

About this page

Tag	`QB1QD9E6VH`
URI	`m2://QB1QD9E6VH`
Kind	proof
Writing	W0 stub
MSC	15A15
Proves	Determinant of a triangular matrix
Uses	Leibniz formula for determinants, Upper triangular matrix, Lower triangular matrix, Sign of a permutation
Changed	4 November 2026

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