Math 2.0

statement proposition

Rule of Sarrus

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.

Explanations

L1 human

The rule of Sarrus is a memory aid for 3×33 \times 3 determinants. Write the first two columns again to the right of the matrix. Add the three products down to the right, and subtract the three products up to the right.

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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

Sources

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

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