### 2.79 Convert a matrix to row echelon form and to reduced row echelon form

Problem: Given a matrix A, convert it to REF and RREF. Below shows how to

convert the matrix A to RREF. To convert to REF (TODO). One reason to convert Matrix $$A$$ to its row echelon form, is to ﬁnd the rank of $$A$$. If matrix $$A$$ is a $$4\times 4$$, and when converted to its row echelon form we ﬁnd that one of the rows is all zeros, then the rank of $$A$$ will be 3 and not full rank.

 Mathematica Remove["Global*"] (mat={{1, 1, -2, 1}, {3, 2, 4, -4}, {4, 3, 3, -4}})//MatrixForm  $\left ( {\begin {array}{cccc} 1 & 1 & -2 & 1 \\ 3 & 2 & 4 & -4 \\ 4 & 3 & 3 & -4 \\ \end {array}} \right )$ MatrixForm[RowReduce[mat]]  $\left ( {\begin {array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & -1 \\ \end {array}} \right )$

 Matlab clear all; A=[1 1 -2 1 3 2 4 -4 4 3 3 -4]; rref(A)  ans = 1 0 0 2 0 1 0 -3 0 0 1 -1 

 Maple A:=Matrix([ [1,1,-2,1],[3,2,4,-4],[4,3,3,-4]]); LinearAlgebra:-ReducedRowEchelonForm(A); ` \left [ {\begin {array}{cccc} 1&0&0&2\\ \noalign {\medskip }0&1&0&-3 \\ \noalign {\medskip }0&0&1&-1\end {array}} \right ]