Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof of the determinant of the inverse

Builds on Invertible matrixDeterminant is multiplicativeDeterminant of the identity matrixUnit of a ring

Proof

  1. Since AA is invertible, there is an n×nn \times n matrix A−1A^{-1} with AA−1=A−1A=InA A^{-1} = A^{-1} A = I_n.
  2. By the product rule, det⁡A⋅det⁡(A−1)=det⁡(AA−1)=det⁡In\det A \cdot \det(A^{-1}) = \det(A A^{-1}) = \det I_n.
  3. By the determinant of the identity, det⁡In=1\det I_n = 1. So det⁡A⋅det⁡(A−1)=1\det A \cdot \det(A^{-1}) = 1.
  4. Since KK is commutative, also det⁡(A−1)⋅det⁡A=1\det(A^{-1}) \cdot \det A = 1.
  5. By the definition of a unit, steps 3 and 4 show that det⁡A\det A is a unit of KK with inverse det⁡(A−1)\det(A^{-1}).
  6. So det⁡(A−1)=(det⁡A)−1\det(A^{-1}) = (\det A)^{-1}.

Idea

The determinant turns products of matrices into products of ring elements and sends InI_n to 11. So the equation AA−1=InA A^{-1} = I_n becomes det⁡A⋅det⁡(A−1)=1\det A \cdot \det(A^{-1}) = 1.

Reading path

To understand this page from the ground up, read these 105 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 the determinant is multiplicative by expanding the rows, 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.

  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. Proposition concept
  15. Transposition concept
  16. Vector space definition
  17. Identity element definition
  18. Set definition
  19. Ring definition
  20. Subgroup definition
  21. Symmetric group definition
  22. Matrix multiplication definition
  23. Square matrix definition
  24. Successor function definition
  25. Conjunction definition
  26. Disjunction definition
  27. Implication definition
  28. Predicate definition
  29. Linear map definition
  30. Inverse element definition
  31. Empty set object
  32. Family of sets definition
  33. Ordered pair definition
  34. Unit of a ring definition
  35. Transposition definition
  36. Proof that an intersection of subgroups is a subgroup proof
  37. Intersection definition
  38. Union definition
  39. Invertible matrix definition
  40. Set-builder notation concept
  41. Addition of natural numbers definition
  42. Biconditional definition
  43. The successor function is injective statement
  44. Zero is not a successor statement
  45. Existential quantifier definition
  46. Universal quantifier definition
  47. Multilinear map definition
  48. Proof of the uniqueness of inverses proof
  49. Cartesian product definition
  50. Disjoint sets definition
  51. Subset definition
  52. Axiom of extensionality statement
  53. Proof of the commutativity of addition by induction proof
  54. Union of a family of sets definition
  55. Order on the natural numbers definition
  56. Uniqueness quantifier definition
  57. Alternating multilinear map definition
  58. Uniqueness of inverses statement
  59. Binary relation definition
  60. Partition of a set definition
  61. Determinant definition
  62. Commutativity of addition statement
  63. Proof of strong induction from ordinary induction proof
  64. Reflexive relation definition
  65. Symmetric relation definition
  66. Transitive relation definition
  67. Proof of the laws of exponents in a group proof
  68. Proof of the subgroup criterion proof
  69. Proof that a matrix with two equal rows has determinant zero proof
  70. Function definition
  71. Proof that the determinant of the identity matrix is one proof
  72. Proof that the determinant is multilinear in the rows proof
  73. Strong induction statement
  74. Equivalence relation definition
  75. Subgroup criterion statement
  76. Determinant of a matrix with two equal rows statement
  77. Laws of exponents in a group statement
  78. Determinant of the identity matrix statement
  79. Determinant is multilinear in the rows statement
  80. Injective function definition
  81. Proof of the well-ordering principle from strong induction proof
  82. Proof that disjoint cycles commute proof
  83. Intersection of subgroups is a subgroup statement
  84. Equivalence class definition
  85. Well-ordering principle statement
  86. Disjoint cycles commute statement
  87. Proof that equivalence classes form a partition proof
  88. Subgroup generated by a subset definition
  89. Proof of the division algorithm proof
  90. Equivalence classes form a partition statement
  91. Division algorithm statement
  92. Proof of the decomposition of a permutation into disjoint cycles proof
  93. Decomposition of a permutation into disjoint cycles statement
  94. Proof of the decomposition of a permutation into transpositions proof
  95. Decomposition of a permutation into transpositions statement
  96. Sign of a permutation definition
  97. Proof that the symmetric group is generated by transpositions proof
  98. The symmetric group is generated by transpositions statement
  99. Proof that permuting rows multiplies the determinant by the sign proof
  100. Permuting rows multiplies the determinant by the sign statement
  101. Proof of the Leibniz formula from the defining properties of the determinant proof
  102. Leibniz formula for determinants statement
  103. Proof that the determinant is multiplicative by expanding the rows proof
  104. Determinant is multiplicative statement
  105. Proof of the determinant of the inverse proof

Where this leads

154 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Invertible_matrix, Other properties
About this page
TagKS54E31XWV
URIm2://KS54E31XWV
Kindproof
WritingW0 stub
MSC15A15
ProvesDeterminant of the inverse
UsesInvertible matrix, Determinant is multiplicative, Determinant of the identity matrix, Unit of a ring
Changed4 November 2026

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