Math 2.0

Linear and multilinear algebra; matrix theory Matrices and matrix algebra Theorem

A matrix is invertible iff its determinant is a unit

Builds on DeterminantInvertible matrixAdjugate matrix

Statement

Let KK be a commutative ring and let AA be an n×nn \times n matrix over KK. Then AA is invertible if and only if its determinant det⁡A\det A is a unit of KK. In particular, over a field, AA is invertible if and only if det⁡A≠0\det A \ne 0.

Remarks

If det⁡A\det A is a unit, then the adjugate gives the inverse A−1=(det⁡A)−1adj⁡(A)A^{-1} = (\det A)^{-1} \operatorname{adj}(A). Over ℤ\Z, the matrix (2001)\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} has determinant 2≠02 \ne 0, but it has no inverse with integer entries. Over a field, this is one of many equivalent conditions for invertibility, with rank⁡A=n\rank A = n and ker⁡A=0\ker A = 0.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionMatrix multiplication, definition, no evidence levelMatrix multiplicationdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionInvertible matrix, definition, no evidence levelInvertible matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionMinor, definition, no evidence levelMinordefinitionCofactor, definition, no evidence levelCofactordefinitionAdjugate matrix, definition, no evidence levelAdjugate matrixdefinitionA matrix is invertible iff its determinant is a unit, statement, no evidence levelA matrix is invertibl…statement
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Matrix multiplication definition
  5. Square matrix definition
  6. Transpose definition
  7. Linear map definition
  8. Invertible matrix definition
  9. Multilinear map definition
  10. Alternating multilinear map definition
  11. Determinant definition
  12. Minor definition
  13. Cofactor definition
  14. Adjugate matrix definition
  15. A matrix is invertible iff its determinant is a unit statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Invertible_matrix, Invertible matrix theorem
About this page
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Kindstatement, theorem
WritingW0 stub
MSC15A15
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UsesDeterminant, Invertible matrix, Adjugate matrix

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