Math 2.0

Linear and multilinear algebra; matrix theory Systems of linear equations Theorem

Cramer's rule

Builds on DeterminantSystem of linear equations

Statement

Let FF be a field, let AA be an n×nn \times n matrix over FF with det⁡A≠0\det A \ne 0, and let b∈Fnb \in F^n. For 1≤i≤n1 \le i \le n, let AiA_i be the matrix that results when column ii of AA is replaced by bb. Then the linear system Ax=bAx = b has exactly one solution x=(x1,…,xn)x = (x_1, \dots, x_n), and xi=det⁡Aidet⁡A(1≤i≤n).x_i = \frac{\det A_i}{\det A} \qquad (1 \le i \le n).

Remarks

For n=2n = 2, the system ax+by=eax + by = e, cx+dy=fcx + dy = f has the solution x=(ed−bf)/(ad−bc)x = (ed - bf)/(ad - bc) and y=(af−ec)/(ad−bc)y = (af - ec)/(ad - bc). The rule follows from the formula for the inverse with the adjugate. It needs n+1n + 1 determinants, so it is slow for large nn compared with Gaussian elimination.

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Identity matrix, object, no evidence levelIdentity matrixobjectLinear equation, definition, no evidence levelLinear equationdefinitionMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSystem of linear equations, definition, no evidence levelSystem of linear equa…definitionSquare 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 levelDeterminantdefinitionCramer's rule, statement, no evidence levelCramer's rulestatement
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  1. Identity matrix object
  2. Linear equation definition
  3. Matrix definition
  4. Vector space definition
  5. System of linear equations definition
  6. Square matrix definition
  7. Linear map definition
  8. Multilinear map definition
  9. Alternating multilinear map definition
  10. Determinant definition
  11. Cramer's rule statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cramer's_rule, General case
About this page
Tag0977KZ6PQB
URIm2://0977KZ6PQB
Kindstatement, theorem
WritingW1 stated
MSC15A06, 15A15
StatusThis copy of the site has no build data. Run m2 status 0977KZ6PQB.
UsesDeterminant, System of linear equations
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