Question about Multiplying Matrices

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Is it ok to reduce the two matrices through row operations first before multiplying them together or will the answer no longer be row equivalent?

Thanks for any input!
 
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Reducing a matrix through row operations is the same as multiplying the matrix by the elementary matrices corresponding to each row operation (the elementary matrix corresponding to a given row operation is the matrix you get by applying that row operation to the identity matrix). That is, if matrix A can be reduced to A' by row operations corresponding to elementary matrices d1, d2... dn and B can be reduced to B' by row operations corresponding to elementary matrices e1, e2, ..., e[sun]m[/sub], then A'= d1d2...dnA and B'= e1e2...emB so that A'B'= d1d2...dnAe1e2...emB which is, generally, NOT the same as AB.
 
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