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 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.

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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?

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.

The statement is false. The echelon form of a matrix is not unique, but the reduced echelon form is unique.

The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. Choose the correct answer below.

The statement is false. The pivot positions in a matrix are determined completely by the positions of the leading entries in the nonzero rows of any echelon form obtained from the matrix.

Reducing a matrix to echelon form is called the forward phase of the row reduction process. Choose the correct answer below.

The statement is true. Reducing a matrix to echelon form is called the forward phase and reducing a matrix to reduced echelon form is called the backward phase.

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Whenever a system has free variables, the solution set contains many solutions. Choose the correct answer below.

The statement is false. The existence of at least one solution is not related to the presence or absence of free variables. If the system is inconsistent, the solution set is empty.

A general solution of a system is an explicit description of all solutions of the system. Choose the correct answer below.

The statement is true. The row reduction algorithm leads directly to an explicit description of the solution set of a linear system when the algorithm is applied to the augmented matrix of the system, leading to a general solution of a system.

Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent

The system is consistent because the rightmost column of the augmented matrix is not a pivot column.

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

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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.

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Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability2nd EditionTom M. Apostol

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