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I am trying to calculate the rank of a matrix and everytime I search for the steps required to calculate the rank of a matrix, the answer always uses the terms row echelon form and reduced row echelon form interchangeably when calculating the rank which is really confusing.

I have read this question row echelon vs reduced row echelon form in which the given answer says that we use row echelon form instead of reduced row echelon form (as it's a tedious process) to calculate the rank however one the answers from a paper my university gave me had this written on it:

Alternatively the rank is obtained by counting number of non-all-zero
rows in reduced matrix form above.
As:
rank of A + nullity of A = dimension of A

It seems to me now that it doesn't really matter which form I use? Is there a difference between using row echelon form to calculate the rank and reduced row echelon form to calculate the rank of a matrix? I'm really confused.

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Row echelon form is simply a matrix such that all nonzero rows are above rows of all zereos, and the leading coefficient is a nonzero row is strictly to the right of the leading coefficient of the row above it.

Reduced row echelon form is a matrix that is in row echelon form but adds the condition that the leading coefficient is the only nonzero element in a column.

It may be tedious to go from row echelon form to reduced row echelon form but there are equivalent by a finite number of steps. Thus since they are equivalently matrix then the rank of the matrix must be the same.

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  • $\begingroup$ Thank you, this was the answer I was looking for. $\endgroup$ – Nubcake May 12 '15 at 14:39

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