Exercise 1.2.1

M={{2,1,4,41/4},{3,3/2,34/5,41/12},{0,2/3,5,35/4}};

R=RowReduce[M]; R//MatrixForm

Do[x[i]=R[[i,4]],{i,1,3}]; {x[1],x[2],x[3]}

Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,3}]]," ",M[[i,4]]],{i,1,3}]

Result check O.K.

Clear[x]

Exercise 1.2.2

M={{-142/35,-284/35,32/5,194/35,134/35},   \
   {-3/2,-3,2,3/2,1/2},                    \
   {24/7,48/7,-4,-18/7,2/7}};              \
   M//MatrixForm

R=RowReduce[M]; R//MatrixForm

R={R[[1]],R[[2]],{0,1,0,0,x[2]},{0,0,0,1,x[4]}}; R//MatrixForm

RR=RowReduce[R]

Do[x[i]=RR[[i,5]],{i,1,4}]; {x[1],x[2],x[3],x[4]}

Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,4}]]," ",M[[i,5]]],{i,1,3}]

Clear[x]

Exercise 1.2.3

M={{0,3,-4,5/3,23/12},         \
   {2,7,4/3,3,41/4},           \
   {1/2,-3,2,13/3,41/12},      \
   {7/6,7/3,-14/9,7,35/4}};
M//MatrixForm

M=M[[{2,1,3,4}]];
M[[1]]/=M[[1,1]];
M[[3]]-=M[[1]]/2; M[[4]]-=7M[[1]]/6;
M//MatrixForm

M[[2]]/=M[[2,2]];
M[[3]]+=19M[[2]]/4;
M[[4]]+=7M[[2]]/4;
M//MatrixForm

M[[3]]/=M[[3,3]];
Do[x[4-i]=Simplify[M[[4-i,5]]-Sum[M[[4-i,j]]x[j],{j,5-i,4}]], \
   {i,1,3}];
S={x[1],x[2],x[3],x[4]}

Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,4}]]," ",M[[i,5]]],{i,1,4}]

Clear[x]

Exercise 1.2.4

M={{3,2,-3,4,5},      \
   {2,7,11,-3,41},    \
   {2,-5,3,5,12},     \
   {6,3,-49,2,34}};
M//MatrixForm

M[[1]]/=M[[1,1]];
M[[2]]-=2M[[1]]; M[[3]]-=2M[[1]]; M[[4]]-=2M[[1]]*3;
M//MatrixForm

M[[2]]/=M[[2,2]];
M[[3]]+=19M[[2]]/3; M[[4]]+=M[[2]];
M//MatrixForm

M[[3]]/=M[[3,3]];
M[[4]]+=692M[[3]]/17;
M//MatrixForm

M[[4]]/=M[[4,4]];
Do[x[4-i]=Simplify[M[[4-i,5]]-Sum[M[[4-i,j]]x[j],{j,5-i,4}]], \
   {i,0,3}];
S={x[1],x[2],x[3],x[4]}

Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,4}]]," ",M[[i,5]]],{i,1,4}]

Clear[x]

Exercise 1.2.5

M={{3,2,-3,4,5},     \
   {2,7,11,-3,41}};
M//MatrixForm

M[[1]]/=M[[1,1]];
M[[2]]-=2M[[1]];
M//MatrixForm

M[[2]]/=M[[2,2]];
Do[x[4-i]=Simplify[M[[4-i,5]]-Sum[M[[4-i,j]]x[j],{j,5-i,4}]], \
   {i,2,3}];
S={x[1],x[2],x[3],x[4]}

Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,4}]]," ",M[[i,5]]],{i,1,2}]

Clear[x]

Exercise 1.2.6

M={{2,3,3,-2,3},     \
   {1,2,1,-1,1},     \
   {4,7,5,-4,12}};
M//MatrixForm

M[[1]]/=M[[1,1]];
M[[2]]-=M[[1]]; M[[3]]-=4M[[1]];
M//MatrixForm

M[[2]]/=M[[2,2]];
M[[3]]-=M[[2]];
M//MatrixForm

No solution (see last row).

Exercise 1.2.10
(This exercise has been inserted here because it needs to be run
prior to any other call to AppendRows in a Mathematica session.)

A={{1,2}}; B={{3,4}};
AppendRows[A,B]

<<LinearAlgebra`Master`;
AppendRows[A,B]

Remove[AppendRows];
AppendRows[A,B]

Exercise 1.2.7

Unprotect[C];
C={{0,0,0,0,0},{0,0,0,0,0},{0,0,0,0,0}};
Do[Do[C[[i,j]]=(i+j)/j,{j,1,5}],{i,1,3}];
K={{1},{1},{1}};
M=AppendRows[C,K]; M//MatrixForm

R=RowReduce[M]; R//MatrixForm

Do[x[4-i]=Simplify[R[[4-i,6]]-Sum[R[[4-i,j]]x[j],{j,5-i,5}]], \
   {i,2,3}];
S={x[1],x[2],x[3],x[4],x[5]}

Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,5}]]," ",M[[i,6]]],{i,1,3}]

Clear[x]

Exercise 1.2.8

C={{0,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,0}};
Do[Do[C[[i,j]]=i/j,{j,1,4}],{i,1,5}];
K={{1},{2},{3},{4},{5}};
M=AppendRows[C,K]; M//MatrixForm

R=RowReduce[M]; R//MatrixForm

x[1]=Simplify[R[[1,5]]-Sum[R[[1,j]]x[j],{j,2,4}]];
S={x[1],x[2],x[3],x[4]}

Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,4}]]," ",M[[i,5]]],{i,1,5}]

Clear[x]

Exercise 1.2.9

C={{1,2,3,4,5,6},   \
   {2,2,3,4,5,6},   \
   {3,3,3,4,5,6},   \
   {4,4,4,4,5,6}};
K={{1!},{2!},{3!},{4!}};
M=AppendRows[C,K]; M//MatrixForm

R=RowReduce[M]; R//MatrixForm

Do[x[i]=R[[i,7]],{i,1,3}]; x[4]=-12-3x[6]/2-5x[5]/4;
S={x[1],x[2],x[3],x[4],x[5],x[6]}

Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,6}]]," ",M[[i,7]]],{i,1,4}]

Exercise 1.2.10 - see after Ex. 1.2.6