Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof of the matrix determinant lemma from Sylvester's determinant theorem

Builds on Invertible matrixDistributivity of matrix multiplicationAssociativity of matrix multiplicationDeterminant is multiplicativeSylvester's determinant theoremLeibniz formula for determinantsIdentity matrix

Proof

We factor out AA and reduce the rest to a 1×11 \times 1 determinant.

  1. Let AA be an invertible n×nn \times n matrix over KK, and let u,v∈Knu, v \in K^n be column vectors. Let w=A−1uw = A^{-1} u.
  2. By distributivity and associativity, A(In+wvT)=A+(AA−1)uvTA (I_n + w v^T) = A + (A A^{-1}) u v^T.
  3. By the definition of the inverse, AA−1=InA A^{-1} = I_n. So A(In+wvT)=A+uvTA (I_n + w v^T) = A + u v^T, with the identity matrix InI_n.
  4. By the product rule, det⁡(A+uvT)=det⁡A⋅det⁡(In+wvT)\det(A + u v^T) = \det A \cdot \det(I_n + w v^T).
  5. The matrix ww is n×1n \times 1 and vTv^T is 1×n1 \times n. By Sylvester's determinant theorem, det⁡(In+wvT)=det⁡(I1+vTw)\det(I_n + w v^T) = \det(I_1 + v^T w).
  6. The matrix I1+vTwI_1 + v^T w is the 1×11 \times 1 matrix with entry 1+vTA−1u1 + v^T A^{-1} u.
  7. By the Leibniz formula for n=1n = 1, which has one term, a 1×11 \times 1 matrix has its entry as determinant. So det⁡(I1+vTw)=1+vTA−1u\det(I_1 + v^T w) = 1 + v^T A^{-1} u.
  8. By steps 4 to 7, det⁡(A+uvT)=(1+vTA−1u)det⁡A\det(A + u v^T) = (1 + v^T A^{-1} u) \det A.

Idea

Write A+uvT=A(I+A−1uvT)A + u v^T = A (I + A^{-1} u v^T). The second factor is the identity plus a column times a row. Sylvester's theorem swaps the order of the column and the row, which turns the n×nn \times n determinant into the number 1+vTA−1u1 + v^T A^{-1} u.

Reading path

To understand this page from the ground up, read these 121 pages in order. The foundations come first. The path takes the proofs Proof of the associativity of matrix multiplication, 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 of the distributivity of matrix multiplication, Proof that the determinant is multiplicative by expanding the rows, Proof that permuting rows multiplies the determinant by the sign, Proof that disjoint cycles commute, Proof of the determinant of a block triangular matrix by the Leibniz formula, Proof that a matrix with two equal rows has determinant zero, 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 Sylvester's determinant theorem by two block factorizations, 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.

  1. Binary operation definition
  2. Cycle notation concept
  3. Definition by recursion concept
  4. Divisibility definition
  5. Element definition
  6. Group concept
  7. Identity matrix object
  8. Mathematical induction concept
  9. Matrix definition
  10. Multilinear map concept
  11. Natural numbers object
  12. Peano axioms concept
  13. Permutation concept
  14. Permutation definition
  15. Proposition concept
  16. Transposition concept
  17. Vector space definition
  18. Identity element definition
  19. Set definition
  20. Group homomorphism definition
  21. Ring definition
  22. Subgroup definition
  23. Symmetric group definition
  24. Block matrix definition
  25. Matrix addition definition
  26. Matrix multiplication definition
  27. Square matrix definition
  28. Transpose definition
  29. Successor function definition
  30. Conjunction definition
  31. Disjunction definition
  32. Implication definition
  33. Predicate definition
  34. Linear map definition
  35. Inverse element definition
  36. Empty set object
  37. Family of sets definition
  38. Ordered pair definition
  39. Transposition definition
  40. Proof that an intersection of subgroups is a subgroup proof
  41. Proof of the associativity of matrix multiplication proof
  42. Intersection definition
  43. Proof of the distributivity of matrix multiplication proof
  44. Union definition
  45. Addition of natural numbers definition
  46. Invertible matrix definition
  47. Set-builder notation concept
  48. The successor function is injective statement
  49. Zero is not a successor statement
  50. Biconditional definition
  51. Existential quantifier definition
  52. Universal quantifier definition
  53. Multilinear map definition
  54. Proof of the uniqueness of inverses proof
  55. Cartesian product definition
  56. Disjoint sets definition
  57. Associativity of matrix multiplication statement
  58. Distributivity of matrix multiplication statement
  59. Subset definition
  60. Union of a family of sets definition
  61. Axiom of extensionality statement
  62. Proof of the commutativity of addition by induction proof
  63. Order on the natural numbers definition
  64. Uniqueness quantifier definition
  65. Alternating multilinear map definition
  66. Uniqueness of inverses statement
  67. Binary relation definition
  68. Partition of a set definition
  69. Commutativity of addition statement
  70. Determinant definition
  71. Proof of strong induction from ordinary induction proof
  72. Reflexive relation definition
  73. Symmetric relation definition
  74. Transitive relation definition
  75. Proof of the laws of exponents in a group proof
  76. Proof of the subgroup criterion proof
  77. Proof that a matrix with two equal rows has determinant zero proof
  78. Proof that the determinant of the identity matrix is one proof
  79. Function definition
  80. Proof that the determinant is multilinear in the rows proof
  81. Strong induction statement
  82. Equivalence relation definition
  83. Laws of exponents in a group statement
  84. Subgroup criterion statement
  85. Determinant of a matrix with two equal rows statement
  86. Determinant of the identity matrix statement
  87. Determinant is multilinear in the rows statement
  88. Injective function definition
  89. Proof of the well-ordering principle from strong induction proof
  90. Proof that disjoint cycles commute proof
  91. Intersection of subgroups is a subgroup statement
  92. Equivalence class definition
  93. Well-ordering principle statement
  94. Disjoint cycles commute statement
  95. Subgroup generated by a subset definition
  96. Proof that equivalence classes form a partition proof
  97. Proof of the division algorithm proof
  98. Equivalence classes form a partition statement
  99. Division algorithm statement
  100. Proof of the decomposition of a permutation into disjoint cycles proof
  101. Decomposition of a permutation into disjoint cycles statement
  102. Proof of the decomposition of a permutation into transpositions proof
  103. Decomposition of a permutation into transpositions statement
  104. Sign of a permutation definition
  105. Proof that the symmetric group is generated by transpositions proof
  106. The symmetric group is generated by transpositions statement
  107. Proof that the sign is a homomorphism proof
  108. Sign is a homomorphism statement
  109. Proof that permuting rows multiplies the determinant by the sign proof
  110. Permuting rows multiplies the determinant by the sign statement
  111. Proof of the Leibniz formula from the defining properties of the determinant proof
  112. Leibniz formula for determinants statement
  113. Proof of the determinant of a block triangular matrix by the Leibniz formula proof
  114. Proof that the determinant is multiplicative by expanding the rows proof
  115. Proof that the determinant of a transpose equals the determinant proof
  116. Determinant of a block triangular matrix statement
  117. Determinant is multiplicative statement
  118. Determinant of a transpose statement
  119. Proof of Sylvester's determinant theorem by two block factorizations proof
  120. Sylvester's determinant theorem statement
  121. Proof of the matrix determinant lemma from Sylvester's determinant theorem proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Matrix_determinant_lemma, Proof
About this page
TagNM2636CNW5
URIm2://NM2636CNW5
Kindproof
WritingW0 stub
MSC15A15
ProvesMatrix determinant lemma
UsesInvertible matrix, Distributivity of matrix multiplication, Associativity of matrix multiplication, Determinant is multiplicative, Sylvester's determinant theorem, Leibniz formula for determinants, Identity matrix
Changed4 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.