(A = {{4,2,-1,0},{5,7,6,3},{2,-5,8,1}})//MatrixForm

(E1 = {{1/4,0,0},{0,1,0},{0,0,1}})//MatrixForm

E1.A//MatrixForm

(E2 = {{1,0,0},{-5,1,0},{0,0,1}})//MatrixForm

E2.E1.A//MatrixForm

(E3= {{1,0,0},{0,1,0},{-2,0,1}})//MatrixForm

E3.E2.E1.A//MatrixForm

(E4= {{1,0,0},{0,2/9,0},{0,0,1}})//MatrixForm

E4.E3.E2.E1.A//MatrixForm

(E5= {{1,-1/2,0},{0,1,0},{0,0,1}})//MatrixForm

E5.E4.E3.E2.E1.A//MatrixForm

(E6= {{1,0,0},{0,1,0},{0,6,1}})//MatrixForm

E6.E5.E4.E3.E2.E1.A//MatrixForm

(E7= {{1,0,0},{0,1,0},{0,0,6/109}})//MatrixForm

E7.E6.E5.E4.E3.E2.E1.A//MatrixForm

(E8= {{1,0,19/18},{0,1,0},{0,0,1}})//MatrixForm

E8.E7.E6.E5.E4.E3.E2.E1.A//MatrixForm

(E9= {{1,0,0},{0,1,-29/18},{0,0,1}})//MatrixForm

E9.E8.E7.E6.E5.E4.E3.E2.E1.A//MatrixForm

A//MatrixForm

E1.A//MatrixForm

(E23 = {{1,0,0},{-5,1,0},{-2,0,1}})//MatrixForm

E3.E2//MatrixForm

E23.E1.A//MatrixForm

E3.E2.E1.A//MatrixForm

E4.E23.E1.A//MatrixForm

(E56 = {{1,-1/2,0},{0,1,0},{0,6,1}})//MatrixForm

E6.E5==E56

E56.E4.E23.E1.A//MatrixForm

E7.E56.E4.E23.E1.A//MatrixForm

(E89= {{1,0,19/18},{0,1,-29/18},{0,0,1}})//MatrixForm

E89.E7.E56.E4.E23.E1.A//MatrixForm

RowReduce[A]//MatrixForm

A//MatrixForm

Ap = (({{4, 2, -1, 0}, {2, -5, 8, 1}, {5, 7, 6, 3}}))

RowReduce[Ap]//MatrixForm

Es = {{1,0,0},{0,0,1},{0,1,0}};

Es.A//MatrixForm

A//MatrixForm

B = {{4,2,-1,0},{8,4,-2,0},{2,-5,8,1}};

B//MatrixForm

F1 = {{1/4,0,0},{0,1,0},{0,0,1}};

F1.B//MatrixForm

F2 = {{1,0,0},{-8,1,0},{0,0,1}};

F2.F1.B//MatrixForm

F3 = {{1,0,0},{0,1,0},{-2,0,1}};

F3.F2.F1.B//MatrixForm

F4 = {{1,0,0},{0,0,1},{0,1,0}};

F4.F3.F2.F1.B//MatrixForm

F5 = {{1,0,0},{0,-1/6,0},{0,0,1}};

F5.F4.F3.F2.F1.B//MatrixForm

F6 = {{1,-1/2,0},{0,1,0},{0,0,1}};

F6.F5.F4.F3.F2.F1.B//MatrixForm