1.2 ROW REDUCTION AND ECHELON FORMS

A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Can such a system have a unique​ solution?

No, it cannot have a unique solution. Because there are more variables than​ equations, there must be at least one free variable. If the linear system is consistent and there is at least one free​ variable, the solution set contains infinitely many solutions. If the linear system is​ inconsistent, there is no solution.

In some​ cases, a matrix may be row reduced to more than one matrix in reduced echelon​ form, using different sequences of row operations. Is this statement true or​ false?

The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix.

The row reduction algorithm applies only to augmented matrices for a linear system. Is this statement true or​ false?

The statement is false. The algorithm applies to any​ matrix, whether or not the matrix is viewed as an augmented matrix for a linear system.

If one row in an echelon form of an augmented matrix is [ 0 0 0 5 0] ​, then the associated linear system is inconsistent. Is this statement true or​ false?

The statement is false. The indicated row corresponds to the equation 5x(4)=0, which does not by itself make the system inconsistent.

Finding a parametric description of the solution set of a linear system is the same as solving the system. Is this statement true or​ false?

The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has at least one solution.

A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix. Is this statement true or​ false?

The statement is true. It is the definition of a basic variable.

A) Suppose a system of linear equations has a 3x5 augmented matrix whose fifth column is not a pivot column. Is the system​ consistent? Why or why​ not?

(Portion of the Existence and Uniqueness Theorem) A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column. That​ is, if and only if an echelon form of the augmented matrix has no row of the form​ [0 ... 0​ b] with b nonzero.

What must be true of a linear system for it to have a unique​ solution?

-The system is consistent. -The system has no free variables.

B) In the augmented matrix described​ above, is the rightmost column a pivot​ column?

No.

C) In the echelon form of the augmented​ matrix, is there a row of the form​ [0 0 0 0​ b] with b​ nonzero?

No.

Suppose a 5x7 coefficient matrix for a system has fivefive pivot columns. Is the system​ consistent? Why or why​ not?

There is a pivot position in each row of the coefficient matrix. The augmented matrix will have eight columns and will not have a row of the form [ 0 0 0 0 0 0 0 1], so the system is consistent.

D) Therefore, by the Existence and Uniqueness​ Theorem, the linear system is

consistent.