Chapter 5 - Determinants, Inverses, and Cramer’s Rule

Section 5.1 - Determinants and Inverses from theAdjoint Formula

Example 5.1.1

A={{0,0},{0,d}};

K={{α,β},{θ,δ}};

A.K//MatrixForm

Commands after Example 5.1.1

(A={{a,b},{c,d}})//MatrixForm

(P=A.K)//MatrixForm

Eqn1=P[[1,1]]==1

Eqn2=P[[1,2]]==0

Eqn3=P[[2,1]]==0

Eqn4=P[[2,2]]==1

Solve[{Eqn1,Eqn2,Eqn3,Eqn4},{α,β,θ,δ}]

(K=(1/(a d-b c)) {{d,-b},{-c,a}})//MatrixForm

Simplify[A.K]//MatrixForm

Simplify[K.A]//MatrixForm

Det[A]

Inverse[A]//MatrixForm

Commands after Example 5.1.3

A={{a,b,c},{d,e,f},{g,h,i}};

K={{p,q,r},{u,v,w},{x,y,z}};

(P=A.K)//MatrixForm

Solns=Simplify[Solve[{P==IdentityMatrix[3]},{p,q,r,u,v,w,x,y,z}]]

K=(Simplify[K/.Solns[[1]]]);

Simplify[A.K]//MatrixForm

Simplify[A.K]//MatrixForm

Det[A]

Example 5.1.6

Solve[{5 x-7 y+2 z-w==-3,9 x+3 y-5 z+8 w==-11,-6 x+y-z+7 w==0,x-4 y-3 z+5 w==6},{x,y,z,w}]

(A={{5,-7,2,-1},{9,3,-5,8},{-6,1,-1,7},{1,-4,-3,5}})//MatrixForm

(AInv=Inverse[A])//MatrixForm

(B={{-3},{-11},{0},{6}})//MatrixForm

AInv.B//MatrixForm

RowReduce[Join[A,B,2]]//MatrixForm

Example 5.1.7

Solve[{5 x-7 y+2 z-w==3,9 x+3 y-5 z+8 w==11,-6 x+y-z+7 w==6,x-4 y-3 z+5 w==-1},{x,y,z,w}]

(BHat={{3},{11},{6},{-1}})//MatrixForm

AInv.BHat//MatrixForm

RowReduce[Join[A,BHat,2]]//MatrixForm

Commands after Example 5.1.7

A={{a,b,c},{d,e,f},{g,h,i}};

Join[A,A[[All,1;;2]],2]//MatrixForm

Det[A]