Linear and multilinear algebra; matrix theory Proof

Proof of the product of a matrix and its adjugate by Laplace expansion

Builds on Adjugate matrix Cofactor Matrix multiplication Laplace expansion along a row Laplace expansion along a column Determinant of a matrix with two equal rows Determinant of a transpose Identity matrix

Proof

We compute each entry of the two products by a Laplace expansion.

Let $A = (a_{kl})$ be an $n \times n$ matrix over $K$ with $n \ge 2$ , and let $C_{kl}$ be its cofactors.
By the definition of the adjugate, $\operatorname{adj}(A)_{lk} = C_{kl}$ .
By the definition of the matrix product, $(A \operatorname{adj}(A))_{ik} = \sum_{j=1}^{n} a_{ij} C_{kj}$ for all $i, k$ .
If $i = k$ , the sum in step 3 is $\det A$ by the Laplace expansion along row $i$ .
Let $i \ne k$ . Let $A'$ be the matrix that results when row $k$ of $A$ is replaced by row $i$ of $A$ .
The cofactor $C'_{kj}$ of $A'$ deletes row $k$ , the only row where $A'$ and $A$ differ. So $C'_{kj} = C_{kj}$ for all $j$ .
Row $k$ of $A'$ has entries $a_{ij}$ . By the Laplace expansion of $A'$ along row $k$ and step 6, $\det A' = \sum_{j=1}^{n} a_{ij} C_{kj}$ .
Rows $i$ and $k$ of $A'$ are equal. By the rule for two equal rows, $\det A' = 0$ .
By steps 3 to 8, $(A \operatorname{adj}(A))_{ik}$ is $\det A$ if $i = k$ and $0$ if $i \ne k$ . So $A \operatorname{adj}(A) = \det(A)\, I_n$ , with the identity matrix $I_n$ .
In the same way, $(\operatorname{adj}(A) A)_{ik} = \sum_{j=1}^{n} C_{ji} a_{jk}$ for all $i, k$ .
If $i = k$ , this sum is $\det A$ by the Laplace expansion along column $i$ .
Let $i \ne k$ . Let $A''$ be the matrix that results when column $i$ of $A$ is replaced by column $k$ of $A$ .
The cofactor $C''_{ji}$ of $A''$ deletes column $i$ , the only column where $A''$ and $A$ differ. So $C''_{ji} = C_{ji}$ for all $j$ .
Column $i$ of $A''$ has entries $a_{jk}$ . By the Laplace expansion of $A''$ along column $i$ , $\det A'' = \sum_{j=1}^{n} a_{jk} C_{ji}$ .
Columns $i$ and $k$ of $A''$ are equal, so the transpose of $A''$ has two equal rows. By the rule for the determinant of a transpose and step 8's rule, $\det A'' = 0$ .
By steps 10 to 15, $\operatorname{adj}(A) A = \det(A)\, I_n$ . Together with step 9, $A \operatorname{adj}(A) = \operatorname{adj}(A) A = \det(A)\, I_n$ .

Idea

An entry of $A \operatorname{adj}(A)$ pairs one row of $A$ with the cofactors of another row. For the same row this is the Laplace expansion of $\det A$ . For different rows it is the expansion of a matrix with two equal rows, which is $0$ .

Reading path

To understand this page from the ground up, read these 116 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 Laplace expansion along a column by transposition, 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 that the determinant of a transpose equals the determinant, 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 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
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
Permutation definition
Proposition concept
Transposition concept
Vector space definition
Identity element definition
Set definition
Group homomorphism definition
Subgroup definition
Symmetric group definition
Matrix multiplication definition
Square matrix definition
Transpose 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
Union definition
Set-builder notation concept
Addition of natural numbers definition
The successor function is injective statement
Zero is not a successor statement
Biconditional definition
Existential quantifier definition
Universal quantifier definition
Multilinear map definition
Proof of the uniqueness of inverses proof
Cartesian product definition
Disjoint sets definition
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
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 laws of exponents in a group proof
Proof of the subgroup criterion 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
Minor definition
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
Cofactor definition
Intersection of subgroups is a subgroup statement
Proof that disjoint cycles commute proof
Equivalence class definition
Well-ordering principle statement
Adjugate matrix definition
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 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 that the determinant of a transpose equals the determinant proof
Determinant of a transpose statement
Laplace expansion along a row statement
Proof of the Laplace expansion along a column by transposition proof
Laplace expansion along a column statement
Proof of the product of a matrix and its adjugate by Laplace expansion proof

Where this leads

184 pages build on this one.

Tag	`Q47MWN8W57`
URI	`m2://Q47MWN8W57`
Kind	proof
Writing	W0 stub
MSC	15A09, 15A15
Proves	Product of a matrix and its adjugate
Uses	Adjugate matrix, Cofactor, Matrix multiplication, Laplace expansion along a row, Laplace expansion along a column, Determinant of a matrix with two equal rows, Determinant of a transpose, Identity matrix
Changed	4 November 2026

Proof of the product of a matrix and its adjugate by Laplace expansion

Proof

Idea

Reading path

Where this leads

Further reading

What this page needs for W1

What links here