 # Row equivalence. What is it exactly? When matrices are row equival Irrerbthist6n 2021-12-20 Answered
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
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${\left[\begin{array}{cccc}1& 0& |& 0\\ -3& 1& |& 0\end{array}\right]}_{{R}_{2}⇒{R}_{2}+3{R}_{1}}$ (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.

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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.

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