Prove or disprove: If two matrices of the same order have the same rank then theymust be row equivalent.

The statement is not necessarily true. Two matrices of the same order having the same rank does not imply that they must be row equivalent.

Here's why:

Row equivalence of two matrices means that one can be obtained from the other by a series of elementary row operations. These operations include swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another row.

The rank of a matrix is the maximum number of linearly independent rows or columns. So, two matrices can have the same rank if they have the same maximum number of linearly independent rows or columns.

However, having the same rank does not mean the matrices are row equivalent. For example, consider the two 2x2 matrices A = [1 2; 0 1] and B = [1 1; 0 1]. Both matrices have rank 2, but they are not row equivalent because you cannot obtain one from the other using elementary row operations.

So, the statement "If two matrices of the same order have the same rank then they must be row equivalent" is disproved.