True or False? -if two syastems of equations have row equivalent augmentedmatric

Question

True or False? -if two syastems of equations have row equivalent augmentedmatrices then they have the same solutions. - A matrix may have more than one echelon form, but the pivotpositions will always be the same. - if an m*n matrix and its columns span R^m then Ax=b isconsistent for all vectors b in R^m. True or False? -if two syastems of equations have row equivalent augmentedmatrices then they have the same solutions. - A matrix may have more than one echelon form, but the pivotpositions will always be the same. - if an m*n matrix and its columns span R^m then Ax=b isconsistent for all vectors b in R^m.

Explanation / Answer

(2) is true. Location of pivot elements is not changed bypassing from one echelon form to another by elementary rowtransformations; indeed if you want to change the location of pivotin a given row, you must use rows above it (because the pivot hasonly zeros beneath it), but they will inevitably introduce non-zerovalues to the left of the pivot. (3) This is true. In this case the rank of augmented matrixcan not be higher than the rank of the matrix itself (m ismaximum), so no inconsistency occurs.

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