Question

1----

. Is the statement​ "Elementary row operations on an augmented matrix never change the solution set of the associated linear​ system" true or​ false? Explain.

A.

​True, because elementary row operations are always applied to an augmented matrix after the solution has been found.

B.

​False, because the elementary row operations make a system inconsistent.

C.

​True, because the elementary row operations replace a system with an equivalent system.

D.

​False, because the elementary row operations augment the number of rows and columns of a matrix.

2---

Indicate whether the statements given in parts​ (a) through​ (d) are true or false and justify the answer.

a. Is the statement​ "Every elementary row operation is​ reversible" true or​ false? Explain.

A.

​False, because only interchanging is a reversible row operation.

B.

​True, because interchanging can be reversed by​ scaling, and scaling can be reversed by replacement.

C.

​False, because only scaling and interchanging are reversible row operations.

D.

​True, because​ replacement, interchanging, and scaling are all reversible.

3----

In parts ​(a) through​ (e) below, mark the statement True or False. Justify each answer.

a. The echelon form of a matrix is unique. Choose the correct answer below.

A.

The statement is true. The echelon form of a matrix is always​ unique, but the reduced echelon form of a matrix might not be unique.

B.

The statement is true. Neither the echelon form nor the reduced echelon form of a matrix are unique. They depend on the row operations performed.

C.

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

D.

The statement is false. Both the echelon form and the reduced echelon form of a matrix are unique. They are the same regardless of the chosen row operations.

Answer 1