Math 2.0

statement proposition

Determinant of the identity matrix

Statement

Let KK be a commutative ring and let n≥1n \ge 1. Then det⁡In=1\det I_n = 1, where InI_n is the identity matrix and det⁡\det is the determinant.

Remarks

This is property 3 of the definition of the determinant. It fixes the scale: the unit cube has volume 11. More generally, the determinant of a diagonal matrix is the product of its diagonal entries.

Explanations

L1 human

The identity matrix leaves every vector where it is. It does not change any volume. So its determinant is 11.

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To read this node, read these 9 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 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 levelDeterminantdefinitionDeterminant of the identity matrix, statement, no evidence levelDeterminant of the id…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. Square matrix definition
  5. Linear map definition
  6. Multilinear map definition
  7. Alternating multilinear map definition
  8. Determinant definition
  9. Determinant of the identity matrix statement

Sources

  1. Web https://en.wikipedia.org/wiki/Determinant, Basic rules

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