Question

Determine whether the statement below is true or false. Justify the answer. The echelon form of a matrix is unique Choose the correct answer below. O A. The statement is false. The echelon form of a matrix is not unique, but the reduced echelon form is unique. O B. The statement is true. Both the echelon form and the reduced echelon form of a matrix are unique. They are the same regardless of the chosen row operations. O C. The statement is false. Neither the echelon form nor the reduced echelon form of a matrix are unique. They depend on the row operations performed. OD. 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. Determine whether the statement below is true or false. Justify the answer. 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. Every pivot position is determined by the positions of the leading entries of a matrix in reduced echelon form. OB. The statement is false. The pivot positions in a matrix depend on the location of the pivot column. O C. 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. OD. 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.

Answer 1

The echelon form of a matrix is not unique, but the reduced echelon form is. The pivot positions in a matrix do not depend on row interchanges during the row reduction process.

Step-by-step explanation:

The statement 'The echelon form of a matrix is unique' is false. The echelon form of a matrix is not unique because different sequences of row operations can lead to different echelon forms. However, the reduced echelon form of a matrix, which is also known as row reduced echelon form (RREF), is unique for a given matrix.

Regarding pivot positions, the statement that the pivot positions in a matrix depend on whether row interchanges are used in the row reduction process is false. Pivot positions are determined completely by the positions of the leading entries in the nonzero rows of any echelon form obtained from the matrix. These pivot positions do not change even when row interchanges are used.

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