Question

Is the statement "Two matrices are row equivalent if they have the same number of rows" true or false? Explain. A. True, because two matrices that are row equivalent have the same number of solutions, which means that they have the same number of rows. B. True, because two matrices are row equivalent if they have the same number of rows and column equivalent if they have the same number of columns. C. False, because if two matrices are row equivalent it means that there exists a sequence of row operations that transforms one matrix to the other. D. False, because if two matrices are row equivalent it means that they have the same number of row solutions.

Answer 1

The statement "Two matrices are row equivalent if they have the same number of rows" is false. Row equivalence is determined by the existence of a sequence of elementary row operations that transforms one matrix into the other, regardless of the number of rows. Options A, B, and D are incorrect. The correct answer is option C.

Step-by-step explanation:

The statement "Two matrices are row equivalent if they have the same number of rows" is false. Row equivalence between matrices is determined by the existence of a sequence of elementary row operations that transforms one matrix into the other. The number of rows in the matrices may or may not be the same. Row equivalence preserves the solution set of linear systems represented by the matrices. Therefore, options A and D are incorrect. Option B is also incorrect because column equivalence relates to the number of columns, not row equivalence.

For example, consider the matrices:

[1 0 0] [2 0 0]

[0 1 0] and [0 2 0]

[0 0 1] [0 0 2]

These matrices have the same number of rows but are not row equivalent, as there are no elementary row operations that can transform one matrix into the other.

Therefore, the correct answer is option C: False, because if two matrices are row equivalent it means that there exists a sequence of row operations that transforms one matrix to the other.

Answer 2

Answer: The answer is (C).

Step-by-step explanation: The given statement is - "Two matrices are row equivalent if they have the same number of rows". We are to explain whether the statement is true or false.

What are row equivalent matrices? The answer to this question is -

Two matrices are said to be row equivalent if one of the matrices can be obtained from the other by applying a number of elementary row operations. Or, we can say two matrices of same order are row equivalent if they have same row space.

Thus, the correct option is (C).