False statement: Two matrices are row equivalent if they have the same number of rows. Understanding Matrix Row Equivalence and the Misconception about the Number of Rows

Is the statement​ Two matrices are row equivalent if they have the same number of​ rows true or​ false? Explain.

​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.

The statement Two matrices are row equivalent if they have the same number of rows is false. Two matrices can have the same number of rows and still not be row equivalent.

Matrices are said to be row equivalent if one can be transformed into the other through a sequence of elementary row operations, which include interchanging two rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another.

For example, consider the following two matrices:

A = [1 2 3; 4 5 6]
B = [2 4 6; 8 10 12]

Both matrices have the same number of rows, but they are not row equivalent, since B can be obtained from A by multiplying each row of A by 2.

Therefore, the statement Two matrices are row equivalent if they have the same number of rows is false.

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