Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Determinant is a coefficient of the characteristic polynomial

Builds on Characteristic polynomialDeterminant

Statement

Let KK be a commutative ring, let n≥1n \ge 1, and let AA be an n×nn \times n matrix over KK. Then the constant coefficient of the characteristic polynomial pA(t)=det⁡(tIn−A)p_A(t) = \det(t I_n - A) is pA(0)=(−1)ndet⁡A,p_A(0) = (-1)^n \det A, where det⁡\det is the determinant.

Remarks

The proof sets t=0t = 0: pA(0)=det⁡(−A)=(−1)ndet⁡Ap_A(0) = \det(-A) = (-1)^n \det A. So, over a field, 00 is an eigenvalue of AA if and only if det⁡A=0\det A = 0.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionCharacteristic polynomial, definition, no evidence levelCharacteristic polyno…definitionDeterminant is a coefficient of the characteristic polynomial, statement, no evidence levelDeterminant is a coef…statement
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Linear map definition
  6. Multilinear map definition
  7. Alternating multilinear map definition
  8. Determinant definition
  9. Characteristic polynomial definition
  10. Determinant is a coefficient of the characteristic polynomial statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Characteristic_polynomial, Properties
About this page
Tag0T0X8GTYYS
URIm2://0T0X8GTYYS
Kindstatement, proposition
WritingW0 stub
MSC15A15, 15A18
StatusThis copy of the site has no build data. Run m2 status 0T0X8GTYYS.
UsesCharacteristic polynomial, Determinant

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