Math 2.0

statement proposition

Determinant is a coefficient of the characteristic polynomial

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.

Explanations

L1 human

The constant term of the characteristic polynomial is the determinant, up to a sign. This follows when we put t=0t = 0. So a matrix with determinant zero has the eigenvalue 00.

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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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Sources

  1. Web https://en.wikipedia.org/wiki/Characteristic_polynomial, Properties

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