A = {{1,1,5,-4},{0,3,2,2},{0,0,-3,1},{0,0,0,1}}

A//MatrixForm

Det[A]

E1 = {{0,0,0,1},{0,1,0,0},{0,0,1,0},{1,0,0,0}}

E1//MatrixForm

A = {{4,2,-1,7},{1,-2,3,-2},{3,-5,6,6},{4,-2,3,9}};

A//MatrixForm

Det[A]

E1 = {{1,0,0,0},{-1/4,1,0,0},{0,0,1,0},{0,0,0,1}};

E1//MatrixForm

E1.A//MatrixForm

Det[E1.A]

E2 = {{1,0,0,0},{0,1,0,0},{-3/4,0,1,0},{0,0,0,1}};

E2.E1.A//MatrixForm

E3 = {{1,0,0,0},{0,1,0,0},{0,0,1,0},{-1,0,0,1}};

E3.E2.E1.A//MatrixForm

E4 = {{1,0,0,0},{0,1,0,0},{0,-13/5,1,0},{0,0,0,1}};

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

E5 = {{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,-8/5,0,1}};

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

E6 = {{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,-12/17,1}};

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

Det[A]

Det[E6.E5.E4.E3.E2.E1.A]

E6//MatrixForm

Inverse[E1]//MatrixForm

Lm = {{1,0,0,0},{1/4,1,0,0},{3/4,13/5,1,0},{1,8/5,12/17,1}};

Lm//MatrixForm

Um = E6.E5.E4.E3.E2.E1.A

Um//MatrixForm

Lm.Um//MatrixForm

A//MatrixForm

F1 = {{0,0,0,1},{0,1,0,0},{0,0,1,0},{1,0,0,0}}

F1//MatrixForm

Det[F1]//MatrixForm

F2 = {{1,0,0,0},{0,0,0,1},{0,0,1,0},{0,1,0,0}}

F2//MatrixForm

Det[F2]//MatrixForm