Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Rule of Sarrus

Builds on Determinant

Statement

Let KK be a commutative ring and let a,b,c,d,e,f,g,h,i∈Ka, b, c, d, e, f, g, h, i \in K. Then det⁡(abcdefghi)=aei+bfg+cdh−ceg−bdi−afh,\det \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = aei + bfg + cdh - ceg - bdi - afh, where det⁡\det is the determinant.

Remarks

Copy the first two columns to the right of the matrix. Then the three positive terms are the products along the diagonals that go down to the right. The three negative terms are products along the diagonals that go up to the right in the same picture. This rule is the Leibniz formula for n=3n = 3; it has no analogue with diagonals for n≥4n \ge 4.

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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 levelDeterminantdefinitionRule of Sarrus, statement, no evidence levelRule of Sarrusstatement
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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. Rule of Sarrus statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Rule_of_Sarrus, lead section
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