How can you determine if two matrices are row-equivalent?

Two matrices are considered row-equivalent if one can be transformed into the other through a series of elementary row operations. These operations include swapping two rows, multiplying a row by a non-zero scalar, or adding a multiple of one row to another row. This definition is fundamental in linear algebra, as row equivalence indicates that the matrices represent the same linear system, allowing for the same solutions.

The row operations effectively preserve the row space and other important properties of the matrices, which means they share the same solution set for the systems they represent. Therefore, identifying row equivalence through these operations is critical in solving linear systems and understanding their geometric interpretations.

In contrast, having the same dimensions does not guarantee that the matrices are row-equivalent, as they may still represent different systems of equations. Similarly, having the same determinant pertains to square matrices and indicates that the matrices are invertible, but this does not imply they are row-equivalent. Lastly, the ability to invert a matrix is related to whether the matrix is full rank, but it does not capture the essence of row equivalence since two non-invertible matrices can still be row-equivalent.