A= {{2,3,-1},{2,10,1},{-5,3,1}};

A//MatrixForm

Det[A]

E1 = {{1,0,0},{-1,1,0},{0,0,1}};

E1.A//MatrixForm

Det[E1]

Det[E1.A]

E0 = {{1/2,0,0},{0,1,0},{0,0,1}};

E0.A//MatrixForm

Det[E0.A]

E1.A//MatrixForm

E2 = {{1,0,0},{0,1,0},{5/2,0,1}};

E2.E1.A//MatrixForm

Det[E2.E1.A]

E3 = {{1,0,0},{0,1,0},{0,-3/2,1}};

E3.E2.E1.A//MatrixForm

(U = E3.E2.E1.A)//MatrixForm

E1//MatrixForm

Inverse[E1]//MatrixForm

E2//MatrixForm

Inverse[E2]//MatrixForm

E3//MatrixForm

Inverse[E3]//MatrixForm

(L = Inverse[E1].Inverse[E2].Inverse[E3])//MatrixForm

L.U//MatrixForm

A==L.U

Inverse[L].{{-3},{5},{0}}//MatrixForm

(8-26/3)/7

Solve[2x1-3*2/21-1*13/3==-3,x1]

Inverse[A].{{-3},{5},{0}}//MatrixForm

Sum[k,{k,1,99}]

100!