(*^
::[	Information =

	"This is a Mathematica Notebook file.  It contains ASCII text, and can be
	transferred by email, ftp, or other text-file transfer utility.  It should
	be read or edited using a copy of Mathematica or MathReader.  If you 
	received this as email, use your mail application or copy/paste to save 
	everything from the line containing (*^ down to the line containing ^*)
	into a plain text file.  On some systems you may have to give the file a 
	name ending with ".ma" to allow Mathematica to recognize it as a Notebook.
	The line below identifies what version of Mathematica created this file,
	but it can be opened using any other version as well.";

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	MacintoshStandardFontEncoding; 
	
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	fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6,  14, "Times"; 
	fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20,  18, "Times"; 
	fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15,  14, "Times"; 
	fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12,  12, "Times"; 
	fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; 
	fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5,  12, "Courier"; 
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	fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5,  12, "Courier"; 
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	fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5,  12, "Courier"; 
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	fontset = leftheader, inactive, L2,  12, "Times"; 
	fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7,  12, "Times"; 
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	fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
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	fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	paletteColors = 128; automaticGrouping; currentKernel; 
]
:[font = input; initialization; preserveAspect]
*)
Exercise 1.2.1
(*
:[font = input; initialization; preserveAspect]
*)
M={{2,1,4,41/4},{3,3/2,34/5,41/12},{0,2/3,5,35/4}};
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
R=RowReduce[M]; R//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, 0, -10597/384}, {0, 1, 0, 8015/64}, 
 
   {0, 0, 1, -1435/96}}]
;[o]
                                   10597
                                 -(-----)
1          0          0             384

                                 8015
                                 ----
0          1          0           64

                                   1435
                                 -(----)
0          0          1             96
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[x[i]=R[[i,4]],{i,1,3}]; {x[1],x[2],x[3]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-10597/384, 8015/64, -1435/96}
;[o]
   10597   8015    1435
{-(-----), ----, -(----)}
    384     64      96
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,3}]]," ",M[[i,4]]],{i,1,3}]
(*
:[font = print; inactive; preserveAspect; endGroup]
41 41
-- --
4  4
41 41
-- --
12 12
35 35
-- --
4  4
:[font = input; initialization; preserveAspect]
*)
Result check O.K.
(*
:[font = input; initialization; preserveAspect]
*)
Clear[x]
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.2.2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
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
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-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}}]
;[o]
  142      284    32       194      134
-(---)   -(---)   --       ---      ---
  35       35     5        35       35

  3                        3        1
-(-)                       -        -
  2      -3       2        2        2

24       48                  18     2
--       --                -(--)    -
7        7        -4         7      7
:[font = input; initialization; preserveAspect; startGroup]
*)
R=RowReduce[M]; R//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 0, 1, 3}, {0, 0, 1, 3/2, 5/2}, 
 
   {0, 0, 0, 0, 0}}]
;[o]


1   2   0   1   3

            3   5
            -   -
0   0   1   2   2



0   0   0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
R={R[[1]],R[[2]],{0,1,0,0,x[2]},{0,0,0,1,x[4]}}; R//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 0, 1, 3}, {0, 0, 1, 3/2, 5/2}, 
 
   {0, 1, 0, 0, x[2]}, {0, 0, 0, 1, x[4]}}]
;[o]


1      2      0      1      3

                     3      5
                     -      -
0      0      1      2      2



0      1      0      0      x[2]



0      0      0      1      x[4]
:[font = input; initialization; preserveAspect; startGroup]
*)
RR=RowReduce[R]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{1, 0, 0, 0, 3 - 2*x[2] - x[4]}, {0, 1, 0, 0, x[2]}, 
 
  {0, 0, 1, 0, 5/2 - (3*x[4])/2}, {0, 0, 0, 1, x[4]}}
;[o]
{{1, 0, 0, 0, 3 - 2 x[2] - x[4]}, {0, 1, 0, 0, x[2]}, 
 
               5   3 x[4]
  {0, 0, 1, 0, - - ------}, {0, 0, 0, 1, x[4]}}
               2     2
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[x[i]=RR[[i,5]],{i,1,4}]; {x[1],x[2],x[3],x[4]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{3 - 2*x[2] - x[4], x[2], 5/2 - (3*x[4])/2, x[4]}
;[o]
                          5   3 x[4]
{3 - 2 x[2] - x[4], x[2], - - ------, x[4]}
                          2     2
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,4}]]," ",M[[i,5]]],{i,1,3}]
(*
:[font = print; inactive; preserveAspect; endGroup]
134 134
--- ---
35  35
1 1
- -
2 2
2 2
- -
7 7
:[font = input; initialization; preserveAspect]
*)
Clear[x]
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.2.3
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
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
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{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}}]
;[o]
                        5       23
                        -       --
0       3       -4      3       12

                4               41
                -               --
2       7       3       3       4

1                       13      41
-                       --      --
2       -3      2       3       12

7       7         14            35
-       -       -(--)           --
6       3         9     7       4
:[font = input; initialization; preserveAspect; startGroup]
*)
M=M[[{2,1,3,4}]];
M[[1]]/=M[[1,1]];
M[[3]]-=M[[1]]/2; M[[4]]-=7M[[1]]/6;
M//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 7/2, 2/3, 3/2, 41/8}, 
 
   {0, 3, -4, 5/3, 23/12}, {0, -19/4, 5/3, 43/12, 41/48}, 
 
   {0, -7/4, -7/3, 21/4, 133/48}}]
;[o]
        7       2       3       41
        -       -       -       --
1       2       3       2       8

                        5       23
                        -       --
0       3       -4      3       12

          19    5       43      41
        -(--)   -       --      --
0         4     3       12      48

          7       7     21      133
        -(-)    -(-)    --      ---
0         4       3     4       48
:[font = input; initialization; preserveAspect; startGroup]
*)
M[[2]]/=M[[2,2]];
M[[3]]+=19M[[2]]/4;
M[[4]]+=7M[[2]]/4;
M//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 7/2, 2/3, 3/2, 41/8}, 
 
   {0, 1, -4/3, 5/9, 23/36}, {0, 0, -14/3, 56/9, 35/9}, 
 
   {0, 0, -14/3, 56/9, 35/9}}]
;[o]
        7       2       3       41
        -       -       -       --
1       2       3       2       8

                  4     5       23
                -(-)    -       --
0       1         3     9       36

                  14    56      35
                -(--)   --      --
0       0         3     9       9

                  14    56      35
                -(--)   --      --
0       0         3     9       9
:[font = input; initialization; preserveAspect; startGroup]
*)
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]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{22/3 - (20*x[4])/3, (-17 + 44*x[4])/36, (-5 + 8*x[4])/6, 
 
  x[4]}
;[o]
 22   20 x[4]  -17 + 44 x[4]  -5 + 8 x[4]
{-- - -------, -------------, -----------, x[4]}
 3       3          36             6
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,4}]]," ",M[[i,5]]],{i,1,4}]
(*
:[font = print; inactive; preserveAspect; endGroup]
41 41
-- --
8  8
23 23
-- --
36 36
  5    5
-(-) -(-)
  6    6
35 35
-- --
9  9
:[font = input; initialization; preserveAspect]
*)
Clear[x]
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.2.4
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M={{3,2,-3,4,5},      \
   {2,7,11,-3,41},    \
   {2,-5,3,5,12},     \
   {6,3,-49,2,34}};
M//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{3, 2, -3, 4, 5}, {2, 7, 11, -3, 41}, 
 
   {2, -5, 3, 5, 12}, {6, 3, -49, 2, 34}}]
;[o]
3     2     -3    4     5

2     7     11    -3    41

2     -5    3     5     12

6     3     -49   2     34
:[font = input; initialization; preserveAspect; startGroup]
*)
M[[1]]/=M[[1,1]];
M[[2]]-=2M[[1]]; M[[3]]-=2M[[1]]; M[[4]]-=2M[[1]]*3;
M//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2/3, -1, 4/3, 5/3}, 
 
   {0, 17/3, 13, -17/3, 113/3}, {0, -19/3, 5, 7/3, 26/3}, 
 
   {0, -1, -43, -6, 24}}]
;[o]
        2               4       5
        -               -       -
1       3       -1      3       3

        17                17    113
        --              -(--)   ---
0       3       13        3      3

          19            7       26
        -(--)           -       --
0         3     5       3       3



0       -1      -43     -6      24
:[font = input; initialization; preserveAspect; startGroup]
*)
M[[2]]/=M[[2,2]];
M[[3]]+=19M[[2]]/3; M[[4]]+=M[[2]];
M//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2/3, -1, 4/3, 5/3}, 
 
   {0, 1, 39/17, -1, 113/17}, {0, 0, 332/17, -4, 863/17}, 
 
   {0, 0, -692/17, -7, 521/17}}]
;[o]
         2                 4        5
         -                 -        -
1        3        -1       3        3

                  39                113
                  --                ---
0        1        17       -1       17

                  332               863
                  ---               ---
0        0        17       -4       17

                    692             521
                  -(---)            ---
0        0          17     -7       17
:[font = input; initialization; preserveAspect; startGroup]
*)
M[[3]]/=M[[3,3]];
M[[4]]+=692M[[3]]/17;
M//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2/3, -1, 4/3, 5/3}, 
 
   {0, 1, 39/17, -1, 113/17}, {0, 0, 1, -17/83, 863/332}, 
 
   {0, 0, 0, -1273/83, 11326/83}}]
;[o]
          2                   4         5
          -                   -         -
1         3         -1        3         3

                    39                  113
                    --                  ---
0         1         17        -1        17

                                17      863
                              -(--)     ---
0         0         1           83      332

                                1273    11326
                              -(----)   -----
0         0         0            83      83
:[font = input; initialization; preserveAspect; startGroup]
*)
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]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{86539/5092, -20535/5092, 3957/5092, -11326/1273}
;[o]
 86539    20535   3957    11326
{-----, -(-----), ----, -(-----)}
 5092     5092    5092    1273
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,4}]]," ",M[[i,5]]],{i,1,4}]
(*
:[font = print; inactive; preserveAspect; endGroup]
5 5
- -
3 3
113 113
--- ---
17  17
863 863
--- ---
332 332
  11326    11326
-(-----) -(-----)
  1273     1273
:[font = input; initialization; preserveAspect]
*)
Clear[x]
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.2.5
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M={{3,2,-3,4,5},     \
   {2,7,11,-3,41}};
M//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{3, 2, -3, 4, 5}, {2, 7, 11, -3, 41}}]
;[o]
3    2    -3   4    5

2    7    11   -3   41
:[font = input; initialization; preserveAspect; startGroup]
*)
M[[1]]/=M[[1,1]];
M[[2]]-=2M[[1]];
M//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2/3, -1, 4/3, 5/3}, 
 
   {0, 17/3, 13, -17/3, 113/3}}]
;[o]
        2               4       5
        -               -       -
1       3       -1      3       3

        17                17    113
        --              -(--)   ---
0       3       13        3      3
:[font = input; initialization; preserveAspect; startGroup]
*)
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]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{(-47 + 43*x[3] - 34*x[4])/17, 113/17 - (39*x[3])/17 + x[4], 
 
  x[3], x[4]}
;[o]
 -47 + 43 x[3] - 34 x[4]  113   39 x[3]
{-----------------------, --- - ------- + x[4], x[3], x[4]}
           17             17      17
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,4}]]," ",M[[i,5]]],{i,1,2}]
(*
:[font = print; inactive; preserveAspect; endGroup]
5 5
- -
3 3
113 113
--- ---
17  17
:[font = input; initialization; preserveAspect]
*)
Clear[x]
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.2.6
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
M={{2,3,3,-2,3},     \
   {1,2,1,-1,1},     \
   {4,7,5,-4,12}};
M//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 3, 3, -2, 3}, {1, 2, 1, -1, 1}, 
 
   {4, 7, 5, -4, 12}}]
;[o]
2    3    3    -2   3

1    2    1    -1   1

4    7    5    -4   12
:[font = input; initialization; preserveAspect; startGroup]
*)
M[[1]]/=M[[1,1]];
M[[2]]-=M[[1]]; M[[3]]-=4M[[1]];
M//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 3/2, 3/2, -1, 3/2}, {0, 1/2, -1/2, 0, -1/2}, 
 
   {0, 1, -1, 0, 6}}]
;[o]
       3      3             3
       -      -             -
1      2      2      -1     2

       1        1             1
       -      -(-)          -(-)
0      2        2    0        2



0      1      -1     0      6
:[font = input; initialization; preserveAspect; startGroup]
*)
M[[2]]/=M[[2,2]];
M[[3]]-=M[[2]];
M//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 3/2, 3/2, -1, 3/2}, {0, 1, -1, 0, -1}, 
 
   {0, 0, 0, 0, 7}}]
;[o]
     3    3         3
     -    -         -
1    2    2    -1   2



0    1    -1   0    -1



0    0    0    0    7
:[font = input; initialization; preserveAspect]
*)
No solution (see last row).
(*
:[font = input; initialization; preserveAspect]
*)
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.)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A={{1,2}}; B={{3,4}};
AppendRows[A,B]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
AppendRows[{{1, 2}}, {{3, 4}}]
;[o]
AppendRows[{{1, 2}}, {{3, 4}}]
:[font = input; initialization; preserveAspect; startGroup]
*)
<<LinearAlgebra`Master`;
AppendRows[A,B]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
AppendRows[{{1, 2}}, {{3, 4}}]
;[o]
AppendRows[{{1, 2}}, {{3, 4}}]
:[font = input; initialization; preserveAspect; startGroup]
*)
Remove[AppendRows];
AppendRows[A,B]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{1, 2, 3, 4}}
;[o]
{{1, 2, 3, 4}}
:[font = input; initialization; preserveAspect]
*)
Exercise 1.2.7
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
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
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 3/2, 4/3, 5/4, 6/5, 1}, 
 
   {3, 2, 5/3, 3/2, 7/5, 1}, {4, 5/2, 2, 7/4, 8/5, 1}}]
;[o]
    3   4   5   6
    -   -   -   -
2   2   3   4   5   1

        5   3   7
        -   -   -
3   2   3   2   5   1

    5       7   8
    -       -   -
4   2   2   4   5   1
:[font = input; initialization; preserveAspect; startGroup]
*)
R=RowReduce[M]; R//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, -1/3, -1/2, -3/5, -1}, 
 
   {0, 1, 4/3, 3/2, 8/5, 2}, {0, 0, 0, 0, 0, 0}}]
;[o]
                1      1      3
              -(-)   -(-)   -(-)
1      0        3      2      5    -1

              4      3      8
              -      -      -
0      1      3      2      5      2



0      0      0      0      0      0
:[font = input; initialization; preserveAspect; startGroup]
*)
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]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{-1 + x[3]/3 + x[4]/2 + (3*x[5])/5, 
 
  2 - (4*x[3])/3 - (3*x[4])/2 - (8*x[5])/5, x[3], x[4], x[5]}
;[o]
      x[3]   x[4]   3 x[5]      4 x[3]   3 x[4]   8 x[5]
{-1 + ---- + ---- + ------, 2 - ------ - ------ - ------, 
       3      2       5           3        2        5
 
  x[3], x[4], x[5]}
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,5}]]," ",M[[i,6]]],{i,1,3}]
(*
:[font = print; inactive; preserveAspect; endGroup]
1 1
1 1
1 1
:[font = input; initialization; preserveAspect]
*)
Clear[x]
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.2.8
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
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
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 1/2, 1/3, 1/4, 1}, {2, 1, 2/3, 1/2, 2}, 
 
   {3, 3/2, 1, 3/4, 3}, {4, 2, 4/3, 1, 4}, 
 
   {5, 5/2, 5/3, 5/4, 5}}]
;[o]
    1   1   1
    -   -   -
1   2   3   4   1

        2   1
        -   -
2   1   3   2   2

    3       3
    -       -
3   2   1   4   3

        4
        -
4   2   3   1   4

    5   5   5
    -   -   -
5   2   3   4   5
:[font = input; initialization; preserveAspect; startGroup]
*)
R=RowReduce[M]; R//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 1/2, 1/3, 1/4, 1}, {0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}]
;[o]
    1   1   1
    -   -   -
1   2   3   4   1



0   0   0   0   0



0   0   0   0   0



0   0   0   0   0



0   0   0   0   0
:[font = input; initialization; preserveAspect; startGroup]
*)
x[1]=Simplify[R[[1,5]]-Sum[R[[1,j]]x[j],{j,2,4}]];
S={x[1],x[2],x[3],x[4]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{1 - x[2]/2 - x[3]/3 - x[4]/4, x[2], x[3], x[4]}
;[o]
     x[2]   x[3]   x[4]
{1 - ---- - ---- - ----, x[2], x[3], x[4]}
      2      3      4
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,4}]]," ",M[[i,5]]],{i,1,5}]
(*
:[font = print; inactive; preserveAspect; endGroup]
1 1
2 2
3 3
4 4
5 5
:[font = input; initialization; preserveAspect]
*)
Clear[x]
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.2.9
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
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
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 3, 4, 5, 6, 1}, {2, 2, 3, 4, 5, 6, 2}, 
 
   {3, 3, 3, 4, 5, 6, 6}, {4, 4, 4, 4, 5, 6, 24}}]
;[o]
1    2    3    4    5    6    1

2    2    3    4    5    6    2

3    3    3    4    5    6    6

4    4    4    4    5    6    24
:[font = input; initialization; preserveAspect; startGroup]
*)
R=RowReduce[M]; R//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 0, 0, 0, 0, 0, 1}, {0, 1, 0, 0, 0, 0, 3}, 
 
   {0, 0, 1, 0, 0, 0, 14}, {0, 0, 0, 1, 5/4, 3/2, -12}}]
;[o]


1     0     0     0     0     0     1



0     1     0     0     0     0     3



0     0     1     0     0     0     14

                        5     3
                        -     -
0     0     0     1     4     2     -12
:[font = input; initialization; preserveAspect; startGroup]
*)
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]}
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{1, 3, 14, -12 - (5*x[5])/4 - (3*x[6])/2, x[5], x[6]}
;[o]
                 5 x[5]   3 x[6]
{1, 3, 14, -12 - ------ - ------, x[5], x[6]}
                   4        2
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[Simplify[Sum[M[[i,j]]x[j],{j,1,6}]]," ",M[[i,7]]],{i,1,4}]
(*
:[font = print; inactive; preserveAspect; endGroup]
1 1
2 2
6 6
24 24
:[font = input; initialization; preserveAspect]
*)
Exercise 1.2.10 - see after Ex. 1.2.6
(*
^*)