# Row equivalence. What is it exactly? When matrices are row equival

Row equivalence. What is it exactly?
When matrices are row equivalent... why is this important? If a matrix like:
$\left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right]$
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 arent
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

einfachmoipf

${\left[\begin{array}{cccc}1& 0& |& 0\\ -3& 1& |& 0\end{array}\right]}_{{R}_{2}⇒{R}_{2}+3{R}_{1}}$ (1)
$\left[\begin{array}{cccc}1& 0& |& 0\\ 0& 1& |& 0\end{array}\right]$ (2)
The above corresponding system of homogeneous equations convey the same information.
x = 0 x = 0 −3x + y = 0 y=0
$⇒$ Elementary row operations do not affect the row space of a matrix. In particular, any two row equivalent matrices have the same row space.
$⇒$ Two matrices in reduced row echelon form have the same row space if and only if they are equal.

Donald Cheek
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.