Math 2.0

statement proposition

Determinant of the inverse

Statement

Let KK be a commutative ring and let AA be an invertible n×nn \times n matrix over KK. Then det⁡A\det A is a unit of KK, and det⁡(A−1)=(det⁡A)−1,\det(A^{-1}) = (\det A)^{-1}, where det⁡\det is the determinant.

Remarks

The proof applies the product rule to AA−1=InA A^{-1} = I_n: det⁡A⋅det⁡(A−1)=1\det A \cdot \det(A^{-1}) = 1. Over a field, this says det⁡(A−1)=1/det⁡A\det(A^{-1}) = 1 / \det A.

Explanations

L1 human

The inverse matrix undoes the matrix, so it must undo its scaling of volume. If AA scales volume by 33, then A−1A^{-1} scales volume by 1/31/3. So the determinant of the inverse is the inverse of the determinant.

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To read this node, read these 12 nodes in this order. The foundations come first, and the list ends with this node.

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 matrixdefinitionLinear 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 levelDeterminantdefinitionDeterminant is multiplicative, statement, no evidence levelDeterminant is multip…statementDeterminant of the inverse, statement, no evidence levelDeterminant of the in…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Matrix multiplication definition
  5. Square matrix definition
  6. Linear map definition
  7. Invertible matrix definition
  8. Multilinear map definition
  9. Alternating multilinear map definition
  10. Determinant definition
  11. Determinant is multiplicative statement
  12. Determinant of the inverse statement

Sources

  1. Web https://en.wikipedia.org/wiki/Invertible_matrix, Other properties

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