Vtyjkyuk

When matrices are row equivalent... why is this important? If a matrix like:

$$beginbmatrix 1 & 0 \ -3 & 1 endbmatrix$$

is row equivalent to the identity matrix (add 3 times the first row to the second), what does that mean exactly? Why is this a concept that we have to know as students of linear algebra? These matrices aren't equal.... they are row-equivalent. Why is this a useful concept to know?

My reason for why the concept of row equivalence is important is that the solutions to the two matrix equations $$Ax=b$$ and $$Bx=b$$ are the same as long as $A$ and $B$ are row equivalent. Often time, you want to reduce an original metric equation $Ax=b$ to an equation $Bx=b$ that is easier to solve, where $B$ is row equivalent to $A$ since row operations do not change the solution set.

$$beginbmatrix 1 & 0 &|&0\ -3 & 1 &|&0endbmatrix_R_2rightarrow R_2+3R_1tag1$$

$$beginbmatrix 1 & 0 &|&0\ 0 & 1 &|&0endbmatrix_ mbox mbox mbox mbox mbox mbox mbox mbox mbox mbox mbox mbox mbox mbox mbox mbox mbox mbox mbox mbox mbox tag2$$

The above corresponding system of homogeneous equations convey the same information.
$$beginmatrixx=0&x=0\-3x+y=0&y=0endmatrix$$

$implies$ Elementary row operations do not affect the row space of a matrix. In particular, any two row equivalent matrices have the same row space.

$implies$ Two matrices in reduced row echelon form have the same row space if and only if they are equal.