Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof that the constant coefficient of the characteristic polynomial is a signed determinant

Builds on Characteristic polynomialLeibniz formula for determinantsEvaluation homomorphismDeterminant of a scalar multiplePolynomial ring

Proof

We evaluate the characteristic polynomial at t=0t = 0.

  1. Let A=(aij)A = (a_{ij}) be an n×nn \times n matrix over KK. Let pA(t)=det⁡(tIn−A)p_A(t) = \det(t I_n - A) be its characteristic polynomial in the polynomial ring K[t]K[t].
  2. Write pA(t)=c0+c1t+⋯+cntnp_A(t) = c_0 + c_1 t + \cdots + c_n t^n with ck∈Kc_k \in K. Then pA(0)=c0p_A(0) = c_0, the constant coefficient.
  3. Let ev⁡0 ⁣:K[t]→K\operatorname{ev}_0 \colon K[t] \to K be the evaluation homomorphism at 00. It maps each entry of tIn−At I_n - A to the same entry of −A-A.
  4. By the Leibniz formula, pA(t)p_A(t) is a sum of products of entries of tIn−At I_n - A, with signs ±1\pm 1.
  5. A ring homomorphism preserves sums, products and signs. So by step 4 and the Leibniz formula for −A-A, pA(0)=ev⁡0(pA)=det⁡(−A)p_A(0) = \operatorname{ev}_0(p_A) = \det(-A).
  6. By the rule for the determinant of a scalar multiple with c=−1c = -1, det⁡(−A)=(−1)ndet⁡A\det(-A) = (-1)^n \det A.
  7. By steps 2, 5 and 6, the constant coefficient of pAp_A is pA(0)=(−1)ndet⁡Ap_A(0) = (-1)^n \det A.

Idea

The constant coefficient of a polynomial is its value at 00. Putting t=0t = 0 in det⁡(tI−A)\det(t I - A) gives det⁡(−A)\det(-A), and the factor −1-1 comes out of each of the nn rows.

Reading path

To understand this page from the ground up, read these 111 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 scaling a row scales the determinant, 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 determinant of a scalar multiple, 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. Scalar multiplication of matrices concept
  16. Transposition concept
  17. Vector space definition
  18. Identity element definition
  19. Set definition
  20. Ring definition
  21. Subgroup definition
  22. Symmetric group definition
  23. Scalar multiplication of matrices definition
  24. Square matrix definition
  25. Successor function definition
  26. Conjunction definition
  27. Disjunction definition
  28. Implication definition
  29. Predicate definition
  30. Linear map definition
  31. Inverse element definition
  32. Empty set object
  33. Family of sets definition
  34. Ordered pair definition
  35. Polynomial ring definition
  36. Ring homomorphism definition
  37. Transposition definition
  38. Proof that an intersection of subgroups is a subgroup proof
  39. Intersection definition
  40. Union definition
  41. Set-builder notation concept
  42. Addition of natural numbers definition
  43. Biconditional definition
  44. The successor function is injective statement
  45. Zero is not a successor statement
  46. Existential quantifier definition
  47. Universal quantifier definition
  48. Multilinear map definition
  49. Proof of the uniqueness of inverses proof
  50. Cartesian product definition
  51. Polynomial evaluation definition
  52. Disjoint sets definition
  53. Subset definition
  54. Axiom of extensionality statement
  55. Proof of the commutativity of addition by induction proof
  56. Union of a family of sets definition
  57. Order on the natural numbers definition
  58. Uniqueness quantifier definition
  59. Alternating multilinear map definition
  60. Uniqueness of inverses statement
  61. Binary relation definition
  62. Evaluation homomorphism definition
  63. Partition of a set definition
  64. Determinant definition
  65. Commutativity of addition statement
  66. Proof of strong induction from ordinary induction proof
  67. Reflexive relation definition
  68. Symmetric relation definition
  69. Transitive relation definition
  70. Proof of the laws of exponents in a group proof
  71. Proof of the subgroup criterion proof
  72. Proof that a matrix with two equal rows has determinant zero proof
  73. Characteristic polynomial definition
  74. Function definition
  75. Proof that the determinant of the identity matrix is one proof
  76. Proof that the determinant is multilinear in the rows proof
  77. Strong induction statement
  78. Equivalence relation definition
  79. Subgroup criterion statement
  80. Laws of exponents in a group statement
  81. Determinant of a matrix with two equal rows statement
  82. Determinant of the identity matrix statement
  83. Determinant is multilinear in the rows statement
  84. Injective function definition
  85. Proof of the well-ordering principle from strong induction proof
  86. Proof that disjoint cycles commute proof
  87. Intersection of subgroups is a subgroup statement
  88. Equivalence class definition
  89. Proof that scaling a row scales the determinant proof
  90. Well-ordering principle statement
  91. Disjoint cycles commute statement
  92. Subgroup generated by a subset definition
  93. Proof that equivalence classes form a partition proof
  94. Scaling a row scales the determinant statement
  95. Proof of the division algorithm proof
  96. Equivalence classes form a partition statement
  97. Division algorithm statement
  98. Proof of the determinant of a scalar multiple proof
  99. Proof of the decomposition of a permutation into disjoint cycles proof
  100. Determinant of a scalar multiple statement
  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 permuting rows multiplies the determinant by the sign proof
  108. Permuting rows multiplies the determinant by the sign statement
  109. Proof of the Leibniz formula from the defining properties of the determinant proof
  110. Leibniz formula for determinants statement
  111. Proof that the constant coefficient of the characteristic polynomial is a signed determinant proof

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Characteristic_polynomial, Properties
About this page
Tag7SYH0YD0VW
URIm2://7SYH0YD0VW
Kindproof
WritingW0 stub
MSC15A15, 15A18
ProvesDeterminant is a coefficient of the characteristic polynomial
UsesCharacteristic polynomial, Leibniz formula for determinants, Evaluation homomorphism, Determinant of a scalar multiple, Polynomial ring
Changed4 November 2026

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