A = {{1,0,2,9},{-4, 6, 9,81},{11,12,13,14},{5,4,3,2}};

Det[A]

A//MatrixForm

Inverse[A]//MatrixForm

Eqns = {w+2y+9z==1,-4w+6x+9y+81z==14,11w+12x+13y+14z==0,5w+4x+3y+2z==-10};

Eqns//MatrixForm

Solve[Eqns,{w,x,y,z}]

B = {{1},{14},{0},{-10}};

Inverse[A].B//MatrixForm

B2= {{10},{-14},{4},{6}};

Inverse[A].B2//MatrixForm

(DM = {{2,0,0,0},{0,-5,0,0},{0,0,-7,0},{0,0,0,10}})//MatrixForm

Det[DM]

(LM = {{2,0,0,0},{8,-5,0,0},{-6,12,-7,0},{2,3,4,10}})//MatrixForm

Det[LM]

(UM = {{2,4,2,8},{0,-5,2,10},{0,0,-7,-100},{0,0,0,10}})//MatrixForm

Det[UM]

A//MatrixForm

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

E1.A//MatrixForm

E2 = {{1,0,0,0},{0,1,0,0},{-11,0,1,0},{0,0,0,1}};

E2.E1.A//MatrixForm

E3 = {{1,0,0,0},{0,1,0,0},{0,0,1,0},{-5,0,0,1}};

E3.E2.E1.A//MatrixForm

E4 = {{1,0,0,0},{0,1,0,0},{0,-2,1,0},{0,0,0,1}};

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

E5 = {{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,-2/3,0,1}};

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

E6 = {{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,-55/(3*43),1}};

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

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

Det[A]

D[x,x]

D[x*x,x]

B = {{0,1,12,-9},{2, 5, -7,1},{-10,-9,-8,-7},{6,7,8,9}};

Det[B]

Det[A.B]

Det[A]*Det[B]

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

Det[E6]

Det[E5]

Det[E4]

Det[E3]

Det[E2]

Det[E1]

10!

9+8+7+6+5+4+3+2+1

Det[3*A]

Det[A]

Det[3*A]/Det[A]

Det[1/2*A]

Det[1/2*A]/Det[A]

Det[Inverse[A]]

Transpose[Minors[A]]//MatrixForm

Det[A]*Inverse[A]//MatrixForm

E8 = {{1,0,0,0},{0,1,0,0},{0,0,0,1},{0,0,1,0}}

E8//MatrixForm

Det[E8]