In parts (a) through (e) below, mark the statement True or False. Justify each a
ID: 2252299 • Letter: I
Question
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. Neither the echelon form nor the reduced echelon form of a matrix are unique. They depend on the row operations performed B. 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 C. 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. D. The statement is false The echelon form of a matrix is not unique, btthe reduced echelon form is unique b. The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. Choose the correct answer below. O A. The statement is true. The pivot positions in a matrix are determined completely by the positions of the leading entries of each row which are dependent on row interchanges O B. 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 ° C. The statement is true. Every pivot position is determined by the positions of the leading entries of a matrix in reduced echelon form. 0 D. The statement is false. The pivot positions in a matrix depend on the location of the pivot column. C. Reducing a matrix to echelon form is called the forward phase of the row reduction process. Choose the correct answer below 0 A. 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 B. The statement is false The forward phase does not depend on whether a matrix is in echelon form or reduced echelon form ° C. The statement is true. The forward phase occurs when a linear system has both basic and free variables, which can only be determined by reducing a matrix to echelon form. 0 D. The statement is false. Reducing a matrix to echelon form is called the backward phase and reducing a matrix to reduced echelon form is called the forward phase d. Whenever a system has free variables, the solution set contains many solutions. Choose the correct answer below. A. The statement is true. If a solution has at least one free variable, there are infinitely many solutions O B. The statement is false. A system with one free variable will have only one solution. Click to select your answerExplanation / Answer
a) False. The echelon form of matrix is not unique but row-echelon form of matrix is unique. Option D.
b) False. The pivot positions are uniquely determined and depends on location of pivot column. Option D.
c) True. Reducing matrix to echelon form is forward phase and reducing matrix to row reduced echelon form is backward phase. Option A.
d) True. If a solution has aleast one free variable then the system has infinitely many solutions as that free variable can take any real number as its value. Option A.