Question

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.

1. Adding a multiple of one column of a square matrix to another column changes only the sign of the determinant.

a. False, adding a multiple of one column to another does not change the value of the determinant.
b. False, adding a multiple of one column to another changes the sign of the determinant.
c. True, adding a multiple of one column to another changes only the sign of the determinant.
d. False, adding a multiple of one column to another changes the value of the determinant by the multiple of the minor.

2. Two matrices are column-equivalent when one matrix can be obtained by performing elementary column operations on the other.

a. False, row-equivalent matrices are matrices that can be obtained from each other by performing elementary column operations on the other.
b. True, column-equivalent matrices are matrices that can be obtained from each other by performing elementary row operations on the other.
c. False, row-equivalent matrices are matrices that can be obtained from each other by performing elementary row operations on the other.
d. True, column-equivalent matrices are matrices that can be obtained from each other by performing elementary column operations on the other.

Answer 1

1) a. False, adding a multiple of one column to another does not change the value of the determinant.

2) d. True, column-equivalent matrices are matrices that can be obtained from each other by performing elementary column operations on the other.

Explanation:

1) If the multiple of one column of a matrix A is added to another to form matrix B then we get: |A| = |B|. Here, the value of the determinant does not change. The correct option is A

a. False, adding a multiple of one column to another does not change the value of the determinant.

2) Two matrices can be column-equivalent when one matrix is changed to the other using a sequence of elementary column operations. Correc option is d.

d. True, column-equivalent matrices are matrices that can be obtained from each other by performing elementary column operations on the other.