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Back to Pivots and their influence on solution spaces

Exercise 4

Avg difficulty 2.5

Determine whether the following statements are true or false and provide a justification for your response.

  1. If the coefficient matrix of a linear system has a pivot in the rightmost column, then the system is inconsistent.

  2. If a linear system has two equations and four variables, then it must be consistent.

  3. If a linear system having four equations and three variables is consistent, then the solution is unique.

  4. Suppose that a linear system has four equations and four variables and that the coefficient matrix has four pivots. Then the linear system is consistent and has a unique solution.

  5. Suppose that a linear system has five equations and three variables and that the coefficient matrix has a pivot position in every column. Then the linear system is consistent and has a unique solution.

Austin, Understanding Linear Algebra, §1.4 Pivots and their influence on solution spaces, Exercise 4

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Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.