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

RowReduce[A]//MatrixForm

e1={{0,1,0,0},{1,0,0,0},{0,0,1,0},{0,0,0,1}}

(e1.A)//MatrixForm

e2={{1/2,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}}

(e2.e1.A)//MatrixForm

e3={{1,0,0,0},{0,1,0,0},{-4,0,1,0},{0,0,0,1}}

(e3.e2.e1.A)//MatrixForm

e4={{1,0,0,0},{0,1,0,0},{0,0,1,0},{-2,0,0,1}}

(e4.e3.e2.e1.A)//MatrixForm

e5={{1,0,0,0},{0,0,1,0},{0,1,0,0},{0,0,0,1}}

(e5.e4.e3.e2.e1.A)//MatrixForm

e6={{1,3/2,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}}

(e6.e5.e4.e3.e2.e1.A)//MatrixForm

e7={{1,0,0,0},{0,1,0,0},{0,-2,1,0},{0,0,0,1}}

(e7.e6.e5.e4.e3.e2.e1.A)//MatrixForm

e8={{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,-7,0,1}}

(e8.e7.e6.e5.e4.e3.e2.e1.A)//MatrixForm

e9={{1,0,0,0},{0,1,0,0},{0,0,0,1},{0,0,1,0}}

(e9.e8.e7.e6.e5.e4.e3.e2.e1.A)//MatrixForm

e10={{1,0,0,0},{0,1,0,0},{0,0,-1/18,0},{0,0,0,1}}

(e10.e9.e8.e7.e6.e5.e4.e3.e2.e1.A)//MatrixForm

e11={{1,0,-4,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}}

(e11.e10.e9.e8.e7.e6.e5.e4.e3.e2.e1.A)//MatrixForm

e12={{1,0,0,0},{0,1,-2,0},{0,0,1,0},{0,0,0,1}}

(e12.e11.e10.e9.e8.e7.e6.e5.e4.e3.e2.e1.A)//MatrixForm

e13={{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1/7}}

(e13.e12.e11.e10.e9.e8.e7.e6.e5.e4.e3.e2.e1.A)//MatrixForm

e14={{1,0,0,-1/2},{0,1,0,0},{0,0,1,0},{0,0,0,1}}

(e14.e13.e12.e11.e10.e9.e8.e7.e6.e5.e4.e3.e2.e1.A)//MatrixForm

e15={{1,0,0,0},{0,1,0,1},{0,0,1,0},{0,0,0,1}}

(e15.e14.e13.e12.e11.e10.e9.e8.e7.e6.e5.e4.e3.e2.e1.A)//MatrixForm

e16={{1,0,0,0},{0,1,0,0},{0,0,1,3/2},{0,0,0,1}}

(e16.e15.e14.e13.e12.e11.e10.e9.e8.e7.e6.e5.e4.e3.e2.e1.A)//MatrixForm

(RowReduce[A])//MatrixForm

A//MatrixForm

A

(B = {{0,2,4,-1,3},{2,-3,2,1,-2},{4,-5,6,-2,6},{2,4,-2,0,-9}})//MatrixForm

(e16.e15.e14.e13.e12.e11.e10.e9.e8.e7.e6.e5.e4.e3.e2.e1.B)//MatrixForm

(Em = e16.e15.e14.e13.e12.e11.e10.e9.e8.e7.e6.e5.e4.e3.e2.e1)//MatrixForm

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

Em.Ap//MatrixForm

Ap.Em//MatrixForm

Inverse[Ap]//MatrixForm

Em//MatrixForm

A//MatrixForm

Ap//MatrixForm

(X = {{(Subscript[x, 1])},{(Subscript[x, 2])},{(Subscript[x, 3])},{(Subscript[x, 4])}})//MatrixForm

(B = {{2},{-1},{3},{1}})//MatrixForm

((Ap.X)//MatrixForm) == (B//MatrixForm)

A//MatrixForm

Inverse[Ap].Ap.X//MatrixForm

Inverse[Ap].B//MatrixForm

RowReduce[A]//MatrixForm