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Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof that the scalar triple product is a determinant

Builds on Cross product Dot product Determinant Laplace expansion along a row Cofactor Determinant of a 2 by 2 matrix

Proof

We expand both sides and compare them.

Let $a = (a_1, a_2, a_3)$ , $b = (b_1, b_2, b_3)$ and $c = (c_1, c_2, c_3)$ . Let $M$ be the $3 \times 3$ matrix with the rows $a$ , $b$ and $c$ .
By the definition of the cross product, $b \times c = (b_2 c_3 - b_3 c_2, \; b_3 c_1 - b_1 c_3, \; b_1 c_2 - b_2 c_1)$ .
By the definition of the dot product, $a \cdot (b \times c) = a_1 (b_2 c_3 - b_3 c_2) + a_2 (b_3 c_1 - b_1 c_3) + a_3 (b_1 c_2 - b_2 c_1).$
By the Laplace expansion along the first row, $\det M = a_1 M_{11} - a_2 M_{12} + a_3 M_{13}$ . Here $M_{1j}$ is the minor: the determinant of $M$ without row $1$ and column $j$ .
By the formula for a $2 \times 2$ determinant, the minors are $M_{11} = b_2 c_3 - b_3 c_2, \quad M_{12} = b_1 c_3 - b_3 c_1, \quad M_{13} = b_1 c_2 - b_2 c_1.$
Thus $\det M = a_1 (b_2 c_3 - b_3 c_2) - a_2 (b_1 c_3 - b_3 c_1) + a_3 (b_1 c_2 - b_2 c_1)$ .
The middle term is $-a_2 (b_1 c_3 - b_3 c_1) = a_2 (b_3 c_1 - b_1 c_3)$ . So the expression in step 6 is the expression in step 3.
Hence $a \cdot (b \times c) = \det M$ , which is the statement.

Idea

The coordinates of $b \times c$ are exactly the signed $2 \times 2$ minors that the Laplace expansion along the first row multiplies by $a_1$ , $a_2$ and $a_3$ . So the dot product with $a$ is this expansion term by term.

Reading path

To understand this page from the ground up, read these 114 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 the Laplace expansion along a row by the Leibniz formula, 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 sign is a homomorphism, Proof that the determinant of the identity matrix is one, Proof of the formula for the determinant of a 2 by 2 matrix, 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 sign of a cycle, 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
Cofactor concept
Cross product concept
Cycle notation concept
Definition by recursion concept
Determinant concept
Divisibility definition
Dot product concept
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
Group homomorphism 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
Intersection definition
Set-builder notation concept
Union definition
Proof that an intersection of subgroups is a subgroup proof
Addition of natural numbers definition
Biconditional definition
Existential quantifier definition
Universal quantifier definition
The successor function is injective statement
Zero is not a successor statement
Multilinear map definition
Cartesian product definition
Disjoint sets definition
Proof of the uniqueness of inverses proof
Union of a family of sets definition
Subset definition
Axiom of extensionality statement
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
Uniqueness quantifier definition
Alternating multilinear map definition
Binary relation definition
Partition of a set definition
Uniqueness of inverses statement
Determinant definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Proof that a matrix with two equal rows has determinant zero proof
Function definition
Proof of the subgroup criterion proof
Proof of the laws of exponents in a group proof
Proof that the determinant of the identity matrix is one proof
Minor definition
Strong induction statement
Proof that the determinant is multilinear in the rows proof
Equivalence relation definition
Determinant of a matrix with two equal rows statement
Injective function definition
Subgroup criterion statement
Determinant is multilinear in the rows statement
Determinant of the identity matrix statement
Laws of exponents in a group statement
Proof of the well-ordering principle from strong induction proof
Cofactor definition
Proof that disjoint cycles commute proof
Equivalence class definition
Intersection of subgroups is a subgroup statement
Well-ordering principle statement
Proof that equivalence classes form a partition proof
Disjoint cycles commute statement
Subgroup generated by a subset definition
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 the sign is a homomorphism proof
Proof of the sign of a cycle proof
Sign of a cycle statement
Sign is a homomorphism 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 Laplace expansion along a row by the Leibniz formula proof
Proof of the formula for the determinant of a 2 by 2 matrix proof
Determinant of a 2 by 2 matrix statement
Laplace expansion along a row statement
Proof that the scalar triple product is a determinant proof

Where this leads

1 page builds on this one.

Scalar triple product

Further reading

Web https://en.wikipedia.org/wiki/Triple_product, Scalar triple product

About this page

Tag	`4MKQFXGPDT`
URI	`m2://4MKQFXGPDT`
Kind	proof
Writing	W0 stub
MSC	15A72
Proves	Scalar triple product
Uses	Cross product, Dot product, Determinant, Laplace expansion along a row, Cofactor, Determinant of a 2 by 2 matrix
Changed	4 November 2026

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