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For a system of linear equations to be consistent, what must hold?

  • A.the rank of the coefficient matrix equals that of the augmented matrix
  • B.the rank of the augmented matrix exceeds that of the coefficient matrix
  • C.the coefficient determinant must be zero
  • D.there must be as many equations as unknowns

Answer: A

If augmenting the matrix raises the rank, one equation contradicts the others and no solution exists. When the ranks agree the system is solvable, uniquely if that rank equals the number of unknowns and otherwise with infinitely many solutions.

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