(*^
::[	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.";

	FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2";

	MacintoshStandardFontEncoding; 
	
	fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8,  24, "Times"; 
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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"; 
	fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5,  12, "Courier"; 
	fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5,  12, "Courier"; 
	fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5,  12, "Courier"; 
	fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5,  12, "Courier"; 
	fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287,  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 = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  10, "Times"; 
	fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7,  12, "Times"; 
	fontset = special3, 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]
*)
$Post:=If[MatrixQ[#],MatrixForm[#],#]&
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.4.1
(*
:[font = input; initialization; preserveAspect]
*)
A=Array[(1/(#1+#2-1))&,{5,5}];
B=Array[(#1+#2-1)&,{5,6}];
Unprotect[C]; C=Table[1/(j+1),{i,5},{j,6}];
(*
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A.B
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{5, 437/60, 287/30, 237/20, 212/15, 197/12}, 
 
   {71/20, 5, 129/20, 79/10, 187/20, 54/5}, 
 
   {197/70, 547/140, 5, 853/140, 503/70, 1159/140}, 
 
   {657/280, 1357/420, 3457/840, 5, 4943/840, 2843/420}, 
 
   {1271/630, 2321/840, 4421/1260, 10721/2520, 5, 
 
    14479/2520}}]
;[o]
        437     287     237     212     197
        ---     ---     ---     ---     ---
5       60      30      20      15      12

71              129     79      187     54
--              ---     --      ---     --
20      5       20      10      20      5

197     547             853     503     1159
---     ---             ---     ---     ----
70      140     5       140     70      140

657     1357    3457            4943    2843
---     ----    ----            ----    ----
280     420     840     5       840     420

1271    2321    4421    10721           14479
----    ----    ----    -----           -----
630     840     1260    2520    5       2520
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A.(B+C)
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{737/120, 362/45, 811/80, 923/75, 1045/72, 
 
    586/35}, {171/40, 329/60, 109/16, 819/100, 1151/120, 
 
    1541/140}, {941/280, 299/70, 2953/560, 2209/350, 
 
    2063/280, 4133/490}, 
 
   {937/336, 1777/504, 4857/1120, 21743/4200, 4343/720, 
 
    2703/392}, {1721/720, 2846/945, 5321/1440, 13871/3150, 
 
    77479/15120, 12904/2205}}]
;[o]
737     362     811     923     1045    586
---     ---     ---     ---     ----    ---
120     45      80      75       72     35

171     329     109     819     1151    1541
---     ---     ---     ---     ----    ----
40      60      16      100     120     140

941     299     2953    2209    2063    4133
---     ---     ----    ----    ----    ----
280     70      560     350     280     490

937     1777    4857    21743   4343    2703
---     ----    ----    -----   ----    ----
336     504     1120    4200    720     392

1721    2846    5321    13871   77479   12904
----    ----    ----    -----   -----   -----
720     945     1440    3150    15120   2205
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A.B+A.C
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{737/120, 362/45, 811/80, 923/75, 1045/72, 
 
    586/35}, {171/40, 329/60, 109/16, 819/100, 1151/120, 
 
    1541/140}, {941/280, 299/70, 2953/560, 2209/350, 
 
    2063/280, 4133/490}, 
 
   {937/336, 1777/504, 4857/1120, 21743/4200, 4343/720, 
 
    2703/392}, {1721/720, 2846/945, 5321/1440, 13871/3150, 
 
    77479/15120, 12904/2205}}]
;[o]
737     362     811     923     1045    586
---     ---     ---     ---     ----    ---
120     45      80      75       72     35

171     329     109     819     1151    1541
---     ---     ---     ---     ----    ----
40      60      16      100     120     140

941     299     2953    2209    2063    4133
---     ---     ----    ----    ----    ----
280     70      560     350     280     490

937     1777    4857    21743   4343    2703
---     ----    ----    -----   ----    ----
336     504     1120    4200    720     392

1721    2846    5321    13871   77479   12904
----    ----    ----    -----   -----   -----
720     945     1440    3150    15120   2205
:[font = input; initialization; preserveAspect]
*)
d)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A.(B-C)
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{463/120, 587/90, 2159/240, 1709/150, 4951/360, 
 
    3379/210}, {113/40, 271/60, 487/80, 761/100, 1093/120, 
 
    1483/140}, {127/56, 124/35, 2647/560, 1028/175, 
 
    1961/280, 398/49}, {457/240, 1057/360, 2617/672, 
 
    20257/4200, 5783/1008, 39059/5880}, 
 
   {921/560, 1901/756, 3721/1120, 8621/2100, 73721/15120, 
 
    16579/2940}}]
;[o]
463     587     2159    1709    4951    3379
---     ---     ----    ----    ----    ----
120     90      240     150     360     210

113     271     487     761     1093    1483
---     ---     ---     ---     ----    ----
40      60      80      100     120     140

127     124     2647    1028    1961    398
---     ---     ----    ----    ----    ---
56      35      560     175     280     49

457     1057    2617    20257   5783    39059
---     ----    ----    -----   ----    -----
240     360     672     4200    1008    5880

921     1901    3721    8621    73721   16579
---     ----    ----    ----    -----   -----
560     756     1120    2100    15120   2940
:[font = input; initialization; preserveAspect]
*)
e)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A.B-A.C
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{463/120, 587/90, 2159/240, 1709/150, 4951/360, 
 
    3379/210}, {113/40, 271/60, 487/80, 761/100, 1093/120, 
 
    1483/140}, {127/56, 124/35, 2647/560, 1028/175, 
 
    1961/280, 398/49}, {457/240, 1057/360, 2617/672, 
 
    20257/4200, 5783/1008, 39059/5880}, 
 
   {921/560, 1901/756, 3721/1120, 8621/2100, 73721/15120, 
 
    16579/2940}}]
;[o]
463     587     2159    1709    4951    3379
---     ---     ----    ----    ----    ----
120     90      240     150     360     210

113     271     487     761     1093    1483
---     ---     ---     ---     ----    ----
40      60      80      100     120     140

127     124     2647    1028    1961    398
---     ---     ----    ----    ----    ---
56      35      560     175     280     49

457     1057    2617    20257   5783    39059
---     ----    ----    -----   ----    -----
240     360     672     4200    1008    5880

921     1901    3721    8621    73721   16579
---     ----    ----    ----    -----   -----
560     756     1120    2100    15120   2940
:[font = input; initialization; preserveAspect]
*)
f)
(*
:[font = input; initialization; preserveAspect]
*)
A.(B.C) : not defined
(*
:[font = input; initialization; preserveAspect]
*)
g)
(*
:[font = input; initialization; preserveAspect]
*)
(A.B).C : not defined
(*
:[font = input; initialization; preserveAspect]
*)
h)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixPower[A,5]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{445816483439/80015040000, 
 
    887363751649/274337280000, 5227474328389/2240421120000, 
 
    32958117976009/17923368960000, 
 
    306965630861131/201637900800000}, 
 
   {887363751649/274337280000, 1352380860767/720135360000, 
 
    9105400283419/6721263360000, 
 
    71761331911877/67212633600000, 
 
    213882509167681/241965480960000}, 
 
   {5227474328389/2240421120000, 
 
    9105400283419/6721263360000, 
 
    2736906678823/2800526400000, 
 
    6902507578753/8961684480000, 
 
    180012828253381/282293061120000}, 
 
   {32958117976009/17923368960000, 
 
    71761331911877/67212633600000, 
 
    6902507578753/8961684480000, 
 
    19040362658809/31365895680000, 
 
    162142911595081/322620641280000}, 
 
   {306965630861131/201637900800000, 
 
    213882509167681/241965480960000, 
 
    180012828253381/282293061120000, 
 
    162142911595081/322620641280000, 
 
    151022043178381/362948221440000}}]
;[o]
445816483439      887363751649      5227474328389
------------      ------------      -------------
80015040000       274337280000      2240421120000
 
   32958117976009    306965630861131
   --------------    ---------------
   17923368960000    201637900800000

887363751649      1352380860767     9105400283419
------------      -------------     -------------
274337280000      720135360000      6721263360000
 
   71761331911877    213882509167681
   --------------    ---------------
   67212633600000    241965480960000

5227474328389     9105400283419     2736906678823
-------------     -------------     -------------
2240421120000     6721263360000     2800526400000
 
   6902507578753     180012828253381
   -------------     ---------------
   8961684480000     282293061120000

32958117976009    71761331911877    6902507578753
--------------    --------------    -------------
17923368960000    67212633600000    8961684480000
 
   19040362658809    162142911595081
   --------------    ---------------
   31365895680000    322620641280000

306965630861131   213882509167681   180012828253381
---------------   ---------------   ---------------
201637900800000   241965480960000   282293061120000
 
   162142911595081   151022043178381
   ---------------   ---------------
   322620641280000   362948221440000
:[font = input; initialization; preserveAspect]
*)
i)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Inverse[A].MatrixPower[A,6]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{445816483439/80015040000, 
 
    887363751649/274337280000, 5227474328389/2240421120000, 
 
    32958117976009/17923368960000, 
 
    306965630861131/201637900800000}, 
 
   {887363751649/274337280000, 1352380860767/720135360000, 
 
    9105400283419/6721263360000, 
 
    71761331911877/67212633600000, 
 
    213882509167681/241965480960000}, 
 
   {5227474328389/2240421120000, 
 
    9105400283419/6721263360000, 
 
    2736906678823/2800526400000, 
 
    6902507578753/8961684480000, 
 
    180012828253381/282293061120000}, 
 
   {32958117976009/17923368960000, 
 
    71761331911877/67212633600000, 
 
    6902507578753/8961684480000, 
 
    19040362658809/31365895680000, 
 
    162142911595081/322620641280000}, 
 
   {306965630861131/201637900800000, 
 
    213882509167681/241965480960000, 
 
    180012828253381/282293061120000, 
 
    162142911595081/322620641280000, 
 
    151022043178381/362948221440000}}]
;[o]
445816483439      887363751649      5227474328389
------------      ------------      -------------
80015040000       274337280000      2240421120000
 
   32958117976009    306965630861131
   --------------    ---------------
   17923368960000    201637900800000

887363751649      1352380860767     9105400283419
------------      -------------     -------------
274337280000      720135360000      6721263360000
 
   71761331911877    213882509167681
   --------------    ---------------
   67212633600000    241965480960000

5227474328389     9105400283419     2736906678823
-------------     -------------     -------------
2240421120000     6721263360000     2800526400000
 
   6902507578753     180012828253381
   -------------     ---------------
   8961684480000     282293061120000

32958117976009    71761331911877    6902507578753
--------------    --------------    -------------
17923368960000    67212633600000    8961684480000
 
   19040362658809    162142911595081
   --------------    ---------------
   31365895680000    322620641280000

306965630861131   213882509167681   180012828253381
---------------   ---------------   ---------------
201637900800000   241965480960000   282293061120000
 
   162142911595081   151022043178381
   ---------------   ---------------
   322620641280000   362948221440000
:[font = input; initialization; preserveAspect]
*)
j)
(*
:[font = input; initialization; preserveAspect]
*)
Id[n_]:=Array[If[#1==#2,1,0]&,{n,n}]
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A+3Id[5]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{4, 1/2, 1/3, 1/4, 1/5}, 
 
   {1/2, 10/3, 1/4, 1/5, 1/6}, {1/3, 1/4, 16/5, 1/6, 1/7}, 
 
   {1/4, 1/5, 1/6, 22/7, 1/8}, {1/5, 1/6, 1/7, 1/8, 28/9}}]
;[o]
     1    1    1    1
     -    -    -    -
4    2    3    4    5

1    10   1    1    1
-    --   -    -    -
2    3    4    5    6

1    1    16   1    1
-    -    --   -    -
3    4    5    6    7

1    1    1    22   1
-    -    -    --   -
4    5    6    7    8

1    1    1    1    28
-    -    -    -    --
5    6    7    8    9
:[font = input; initialization; preserveAspect]
*)
k)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Id[5].A
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 1/2, 1/3, 1/4, 1/5}, 
 
   {1/2, 1/3, 1/4, 1/5, 1/6}, {1/3, 1/4, 1/5, 1/6, 1/7}, 
 
   {1/4, 1/5, 1/6, 1/7, 1/8}, {1/5, 1/6, 1/7, 1/8, 1/9}}]
;[o]
    1   1   1   1
    -   -   -   -
1   2   3   4   5

1   1   1   1   1
-   -   -   -   -
2   3   4   5   6

1   1   1   1   1
-   -   -   -   -
3   4   5   6   7

1   1   1   1   1
-   -   -   -   -
4   5   6   7   8

1   1   1   1   1
-   -   -   -   -
5   6   7   8   9
:[font = input; initialization; preserveAspect]
*)
l)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixPower[A,3]-3A+Id[5]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{381/1400, -7961/43200, -55163/1058400, 
 
    -10051/2822400, 560191/31752000}, 
 
   {-7961/43200, 173599/226800, -417607/2116800, 
 
    -1737103/10584000, -5295659/38102400}, 
 
   {-55163/1058400, -417607/2116800, 78359/98000, 
 
    -781801/4233600, -7435559/44452800}, 
 
   {-10051/2822400, -1737103/10584000, -781801/4233600, 
 
    1350683/1646400, -8567459/50803200}, 
 
   {560191/31752000, -5295659/38102400, -7435559/44452800, 
 
    -8567459/50803200, 47880541/57153600}}]
;[o]
381             7961           55163         10051
----          -(-----)      -(-------)    -(-------)
1400            43200         1058400       2822400
 
    560191
   --------
   31752000

  7961        173599          417607        1737103
-(-----)      ------        -(-------)    -(--------)
  43200       226800          2116800       10584000
 
     5295659
   -(--------)
     38102400

   55163        417607      78359           781801
-(-------)    -(-------)    -----         -(-------)
  1058400       2116800     98000           4233600
 
     7435559
   -(--------)
     44452800

   10051        1737103       781801      1350683
-(-------)    -(--------)   -(-------)    -------
  2822400       10584000      4233600     1646400
 
     8567459
   -(--------)
     50803200

 560191         5295659       7435559       8567459
--------      -(--------)   -(--------)   -(--------)
31752000        38102400      44452800      50803200
 
   47880541
   --------
   57153600
:[font = input; initialization; preserveAspect]
*)
Exercise 1.4.2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
L=Array[Min,{2,2}];
M=Array[Max,{2,2}];
M.L-L.M
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 1}, {-1, 0}}]
;[o]
0    1

-1   0
:[font = input; initialization; preserveAspect; startGroup]
*)
L=Array[Min,{3,3}];
M=Array[Max,{3,3}];
M.L-L.M
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 4, 5}, {-4, 0, 0}, {-5, 0, 0}}]
;[o]
0    4    5

-4   0    0

-5   0    0
:[font = input; initialization; preserveAspect; startGroup]
*)
L=Array[Min,{10,10}];
M=Array[Max,{10,10}];
M.L-L.M
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 53, 103, 149, 190, 225, 253, 273, 284, 285}, 
 
   {-53, 0, 49, 93, 131, 162, 185, 199, 203, 196}, 
 
   {-103, -49, 0, 43, 79, 107, 126, 135, 133, 119}, 
 
   {-149, -93, -43, 0, 35, 61, 77, 82, 75, 55}, 
 
   {-190, -131, -79, -35, 0, 25, 39, 41, 30, 5}, 
 
   {-225, -162, -107, -61, -25, 0, 13, 13, -1, -30}, 
 
   {-253, -185, -126, -77, -39, -13, 0, -1, -17, -49}, 
 
   {-273, -199, -135, -82, -41, -13, 1, 0, -17, -51}, 
 
   {-284, -203, -133, -75, -30, 1, 17, 17, 0, -35}, 
 
   {-285, -196, -119, -55, -5, 30, 49, 51, 35, 0}}]
;[o]
0      53     103    149    190    225    253    273
 
   284    285

-53    0      49     93     131    162    185    199
 
   203    196

-103   -49    0      43     79     107    126    135
 
   133    119

-149   -93    -43    0      35     61     77     82
 
   75     55

-190   -131   -79    -35    0      25     39     41
 
   30     5

-225   -162   -107   -61    -25    0      13     13
 
   -1     -30

-253   -185   -126   -77    -39    -13    0      -1
 
   -17    -49

-273   -199   -135   -82    -41    -13    1      0
 
   -17    -51

-284   -203   -133   -75    -30    1      17     17
 
   0      -35

-285   -196   -119   -55    -5     30     49     51
 
   35     0
:[font = input; initialization; preserveAspect; startGroup]
*)
L=Array[Min,{11,11}];
M=Array[Max,{11,11}];
M.L-L.M
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 64, 125, 182, 234, 280, 319, 350, 372, 384, 
 
    385}, {-64, 0, 60, 115, 164, 206, 240, 265, 280, 284, 
 
    276}, {-125, -60, 0, 54, 101, 140, 170, 190, 199, 196, 
 
    180}, {-182, -115, -54, 0, 46, 83, 110, 126, 130, 121, 
 
    98}, {-234, -164, -101, -46, 0, 36, 61, 74, 74, 60, 31}, 
 
   {-280, -206, -140, -83, -36, 0, 24, 35, 32, 14, -20}, 
 
   {-319, -240, -170, -110, -61, -24, 0, 10, 5, -16, -54}, 
 
   {-350, -265, -190, -126, -74, -35, -10, 0, -6, -29, -70}, 
 
   {-372, -280, -199, -130, -74, -32, -5, 6, 0, -24, -67}, 
 
   {-384, -284, -196, -121, -60, -14, 16, 29, 24, 0, -44}, 
 
   {-385, -276, -180, -98, -31, 20, 54, 70, 67, 44, 0}}]
;[o]
0      64     125    182    234    280    319    350
 
   372    384    385

-64    0      60     115    164    206    240    265
 
   280    284    276

-125   -60    0      54     101    140    170    190
 
   199    196    180

-182   -115   -54    0      46     83     110    126
 
   130    121    98

-234   -164   -101   -46    0      36     61     74
 
   74     60     31

-280   -206   -140   -83    -36    0      24     35
 
   32     14     -20

-319   -240   -170   -110   -61    -24    0      10
 
   5      -16    -54

-350   -265   -190   -126   -74    -35    -10    0
 
   -6     -29    -70

-372   -280   -199   -130   -74    -32    -5     6
 
   0      -24    -67

-384   -284   -196   -121   -60    -14    16     29
 
   24     0      -44

-385   -276   -180   -98    -31    20     54     70
 
   67     44     0
:[font = input; initialization; preserveAspect]
*)
Exercise 1.4.3
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
g[i_,j_]=i^2-j^2;
A=Array[g,{3,3}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, -3, -8}, {3, 0, -5}, {8, 5, 0}}]
;[o]
0    -3   -8

3    0    -5

8    5    0
:[font = input; initialization; preserveAspect; startGroup]
*)
p[A]=MatrixPower[A,3]+98A
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}]
;[o]
0   0   0

0   0   0

0   0   0
:[font = input; initialization; preserveAspect]
*)
If A were invertible, then MatrixPower[A,3]+98A = 0 ==>
A.(MatrixPower[A,2]+98Id) = 0 ==>
Inverse[A].A.(MatrixPower[A,2]+98Id) = Inverse[A].0 ==>
MatrixPower[A,2]+98Id = 0, which is not true:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixPower[A,2]+98Id[3]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{25, -40, 15}, {-40, 64, -24}, {15, -24, 9}}]
;[o]
25    -40   15

-40   64    -24

15    -24   9
:[font = input; initialization; preserveAspect]
*)
Now, see what error message Mathematica gives:
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Inverse[A]
(*
:[font = message; inactive; preserveAspect]
LinearSolve::nosol: 
   Linear equation encountered which has no solution.
:[font = output; output; inactive; preserveAspect; endGroup]
Inverse[{{0, -3, -8}, {3, 0, -5}, {8, 5, 0}}]
;[o]
Inverse[{{0, -3, -8}, {3, 0, -5}, {8, 5, 0}}]
:[font = input; initialization; preserveAspect]
*)
Exercise 1.4.4
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[Max,{5,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{1, 2, 3, 4, 5}, {2, 2, 3, 4, 5}, 
 
   {3, 3, 3, 4, 5}, {4, 4, 4, 4, 5}, {5, 5, 5, 5, 5}}]
;[o]
1   2   3   4   5

2   2   3   4   5

3   3   3   4   5

4   4   4   4   5

5   5   5   5   5
:[font = input; initialization; preserveAspect; startGroup]
*)
p[x]=Det[Id[5]x-A]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
-5 - 39*x - 98*x^2 - 85*x^3 - 15*x^4 + x^5
;[o]
                2       3       4    5
-5 - 39 x - 98 x  - 85 x  - 15 x  + x
:[font = input; initialization; preserveAspect; startGroup]
*)
-5Id[5]-39A-98A.A-85A.A.A-15A.A.A.A+A.A.A.A.A
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}]
;[o]
0   0   0   0   0

0   0   0   0   0

0   0   0   0   0

0   0   0   0   0

0   0   0   0   0
:[font = input; initialization; preserveAspect]
*)
From p[A]=0 we have A.(-39Id-98A-85A.A-15A.A.A+A.A.A.A)=5Id
    ==> (-39Id-98A-85A.A-15A.A.A+A.A.A.A)/5 = Inverse[A]
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
(-39Id[5]-98A-85A.A-15A.A.A+A.A.A.A)/5
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-1, 1, 0, 0, 0}, {1, -2, 1, 0, 0}, 
 
   {0, 1, -2, 1, 0}, {0, 0, 1, -2, 1}, {0, 0, 0, 1, -4/5}}]
;[o]


-1     1      0      0      0



1      -2     1      0      0



0      1      -2     1      0



0      0      1      -2     1

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


-1     1      0      0      0



1      -2     1      0      0



0      1      -2     1      0



0      0      1      -2     1

                              4
                            -(-)
0      0      0      1        5
:[font = input; initialization; preserveAspect]
*)
Exercise 1.4.5
(*
:[font = input; initialization; preserveAspect]
*)
A:=Array[Random[Integer,{-10,10}]&,{4,4}];
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Do[Print[Inverse[A]//MatrixForm],{i,1,10}]
(*
:[font = print; inactive; preserveAspect; endGroup]
  453       430       101      89
-(----)   -(----)   -(----)   ----
  3001      3001      3001    3001

217         218     214         248
----      -(----)   ----      -(----)
3001        3001    3001        3001

180         167     219       223
----      -(----)   ----      ----
3001        3001    3001      3001

675       124        71        86
----      ----      ----      ----
3001      3001      3001      3001
  231       1102     65       206
-(----)   -(----)   ----      ----
  6995      6995    1399      1399

 38       151       104         230
----      ----      ----      -(----)
1399      1399      1399        1399

  464        3       70         101
-(----)   -(----)   ----      -(----)
  6995      6995    1399        1399

461         193       160      31
----      -(----)   -(----)   ----
6995        6995      1399    1399
889         480     168       1067
----      -(----)   ----      ----
6001        6001    6001      6001

   25      54         619     105
-(----)   ----      -(----)   ----
  6001    6001        6001    6001

  332        3        299       1006
-(----)   -(----)   -(----)   -(----)
  6001      6001      6001      6001

  131     523       117         650
-(----)   ----      ----      -(----)
  6001    6001      6001        6001
168       383        17       661
----      ----      ----      ----
3331      3331      6662      6662

334        8        235       320
----      ----      ----      ----
3331      3331      3331      3331

440       210       679       145
----      ----      ----      ----
3331      3331      6662      6662

185       164         179       102
----      ----      -(----)   -(----)
3331      3331        3331      3331
189       266         510     549
----      ----      -(----)   ----
2029      2029        2029    2029

  263       295     291        9
-(----)   -(----)   ----      ----
  2029      2029    2029      2029

   67      56       213          98
-(----)   ----      ----      -(----)
  2029    2029      2029        2029

  640       525     793         603
-(----)   -(----)   ----      -(----)
  2029      2029    2029        2029
  307     153       9         1007
-(----)   ---       --        ----
  3010    430       43        3010

  1         1                 1
-(--)     -(--)               --
  70        10      0         70

117         38        7         222
----      -(---)    -(--)     -(----)
1505        215       43        1505

4           1                   4
--        -(-)                -(--)
35          5       0           35
155        10       41       97
---      -(--)    -(--)    -(---)
264        33       66       528

27         1        7        1
--       -(--)    -(--)    -(--)
44         11       11       88

  5               1          1
-(-)              -        -(--)
  8      0        2          16

5          2        3        1
--       -(--)    -(--)    -(--)
22         11       11       44
26         74       127          3
---       ----      ----      -(----)
245       1715      1715        1715

1         214        3        186
--        ----      ---       ----
49        1715      343       1715

19        137        8           38
---       ----      ----      -(----)
245       1715      1715        1715

 8           6      171         212
---       -(---)    ----      -(----)
245         343     1715        1715
263         141     153        46
----      -(----)   ----      ----
4154        2077    8308      2077

   33        6      581         219
-(----)   -(----)   ----      -(----)
  4154      2077    8308        2077

  154        56        29      33
-(----)   -(----)   -(----)   ----
  2077      2077      2077    2077

219         149       457       246
----      -(----)   -(----)   -(----)
4154        2077      8308      2077
361          73     394         353
----      -(----)   ----      -(----)
1001        1001    1001        1001

8            5      11        1
--        -(---)    --        --
91          182     91        91

271       115       293         387
----      ----      ----      -(----)
1001      2002      1001        1001

  27       9          38      36
-(--)     ---       -(--)     --
  77      154         77      77
:[font = input; initialization; preserveAspect]
*)
The probability for random integer matrices (in a given
range) to be invertble is quite high, as our example
shows. For matrices of real numbers ("true reals", as 
opposed to their computer representations), that proba-
bility actually equals 1.
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.4.6
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[a,{3,3}];
B=Array[b,{3,3}];
C=Array[c,{3,3}];
lhs=(A+B).C;
rhs=A.C+B.C;
Simplify[lhs-rhs]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}]
;[o]
0   0   0

0   0   0

0   0   0
:[font = input; initialization; preserveAspect]
*)
Exercise 1.4.7
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[a,{5,5}];
lhs=(k*Id[5]).A;
mhs=k*A;
rhs=A.(k*Id[5]);
lhs-mhs
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}]
;[o]
0   0   0   0   0

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]
*)
rhs-mhs
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}]
;[o]
0   0   0   0   0

0   0   0   0   0

0   0   0   0   0

0   0   0   0   0

0   0   0   0   0
:[font = input; initialization; preserveAspect]
*)
Exercise 1.4.8
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Myrand:=Random[Integer,{-100,-1}];
Table[Myrand/Myrand,{i,5},{j,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{61/33, 1/3, 82/75, 8/11, 23/24}, 
 
   {8, 4/7, 29/3, 55/18, 1}, 
 
   {20/9, 91/92, 31/12, 43/8, 53/35}, 
 
   {5/12, 61/19, 32/19, 3/59, 13/3}, 
 
   {92/49, 98/93, 44/35, 29/28, 79/52}}]
;[o]
61   1    82   8    23
--   -    --   --   --
33   3    75   11   24

     4    29   55
     -    --   --
8    7    3    18   1

20   91   31   43   53
--   --   --   --   --
9    92   12   8    35

5    61   32   3    13
--   --   --   --   --
12   19   19   59   3

92   98   44   29   79
--   --   --   --   --
49   93   35   28   52
:[font = input; initialization; preserveAspect]
*)
Exercise 1.4.9
a)
(*
:[font = input; initialization; preserveAspect]
*)
f:=Function[{i,j},If[i<=j,Myrand,0]]
(*
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
L=Array[f,{6,6}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-69, -94, -10, -98, -17, -82}, 
 
   {0, -49, -97, -4, -59, -10}, {0, 0, -5, -42, -45, -84}, 
 
   {0, 0, 0, -87, -74, -85}, {0, 0, 0, 0, -97, -76}, 
 
   {0, 0, 0, 0, 0, -43}}]
;[o]
-69   -94   -10   -98   -17   -82

0     -49   -97   -4    -59   -10

0     0     -5    -42   -45   -84

0     0     0     -87   -74   -85

0     0     0     0     -97   -76

0     0     0     0     0     -43
:[font = input; initialization; preserveAspect; startGroup]
*)
M=Array[f,{6,6}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-3, -35, -20, -34, -57, -45}, 
 
   {0, -43, -5, -87, -95, -88}, {0, 0, -66, -35, -39, -95}, 
 
   {0, 0, 0, -33, -49, -40}, {0, 0, 0, 0, -19, -16}, 
 
   {0, 0, 0, 0, 0, -31}}]
;[o]
-3    -35   -20   -34   -57   -45

0     -43   -5    -87   -95   -88

0     0     -66   -35   -39   -95

0     0     0     -33   -49   -40

0     0     0     0     -19   -16

0     0     0     0     0     -31
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
L.M
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{207, 6457, 2510, 14108, 18378, 19061}, 
 
   {0, 2107, 6647, 7790, 9755, 14941}, 
 
   {0, 0, 330, 1561, 3108, 5479}, 
 
   {0, 0, 0, 2871, 5669, 7299}, {0, 0, 0, 0, 1843, 3908}, 
 
   {0, 0, 0, 0, 0, 1333}}]
;[o]
207     6457    2510    14108   18378   19061

0       2107    6647    7790    9755    14941

0       0       330     1561    3108    5479

0       0       0       2871    5669    7299

0       0       0       0       1843    3908

0       0       0       0       0       1333
:[font = input; initialization; preserveAspect]
*)
Exercise 1.4.10
(*
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect]
*)
g:=Function[{i,j},If[i<j,Myrand,0]]
(*
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[g,{5,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, -33, -66, -82, -17}, {0, 0, -61, -62, -20}, 
 
   {0, 0, 0, -79, -100}, {0, 0, 0, 0, -10}, {0, 0, 0, 0, 0}}]
;[o]
0      -33    -66    -82    -17

0      0      -61    -62    -20

0      0      0      -79    -100

0      0      0      0      -10

0      0      0      0      0
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixPower[A,5]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}]
;[o]
0   0   0   0   0

0   0   0   0   0

0   0   0   0   0

0   0   0   0   0

0   0   0   0   0
:[font = input; initialization; preserveAspect]
*)
d)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A.A.A
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, -159027, -273900}, 
 
   {0, 0, 0, 0, -48190}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0}}]
;[o]
0         0         0         -159027   -273900

0         0         0         0         -48190

0         0         0         0         0

0         0         0         0         0

0         0         0         0         0
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixPower[A,4]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0, 1590270}, {0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}]
;[o]
0         0         0         0         1590270

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]
*)
A=Array[g,{6,6}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, -42, -5, -69, -92, -40}, 
 
   {0, 0, -6, -85, -13, -73}, {0, 0, 0, -58, -31, -24}, 
 
   {0, 0, 0, 0, -24, -23}, {0, 0, 0, 0, 0, -62}, 
 
   {0, 0, 0, 0, 0, 0}}]
;[o]
0     -42   -5    -69   -92   -40

0     0     -6    -85   -13   -73

0     0     0     -58   -31   -24

0     0     0     0     -24   -23

0     0     0     0     0     -62

0     0     0     0     0     0
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixPower[A,3]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, -14616, -100452, -240962}, 
 
   {0, 0, 0, 0, -8352, -146016}, {0, 0, 0, 0, 0, -86304}, 
 
   {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0}}]
;[o]
0         0         0         -14616    -100452   -240962

0         0         0         0         -8352     -146016

0         0         0         0         0         -86304

0         0         0         0         0         0

0         0         0         0         0         0

0         0         0         0         0         0
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixPower[A,6]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}}]
;[o]
0   0   0   0   0   0

0   0   0   0   0   0

0   0   0   0   0   0

0   0   0   0   0   0

0   0   0   0   0   0

0   0   0   0   0   0
:[font = input; initialization; preserveAspect]
*)
We observe that, if A is a strictly upper triangular 
n*n matrix, MatrixPower[n,i] will have exactly n-i
nonzero lines/columns.
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.4.11
(*
:[font = input; initialization; preserveAspect]
*)
From our experiments in exercise 10, we can conjecture
that a strictly upper triangular n*n matrix will be
nilpotent of index <= n, since MatrixPower[A,n-1] can
have one nonzero element in the upper right corner,
while MatrixPower[A,n] = 0.
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.4.12
(*
:[font = input; initialization; preserveAspect]
*)
ut:=Function[{i,j},If[i<j,x[10i+j],0]]
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
T=Array[ut,{5,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, x[12], x[13], x[14], x[15]}, 
 
   {0, 0, x[23], x[24], x[25]}, {0, 0, 0, x[34], x[35]}, 
 
   {0, 0, 0, 0, x[45]}, {0, 0, 0, 0, 0}}]
;[o]
0       x[12]   x[13]   x[14]   x[15]

0       0       x[23]   x[24]   x[25]

0       0       0       x[34]   x[35]

0       0       0       0       x[45]

0       0       0       0       0
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
Clear[$Post]
T.T
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{0, 0, x[12]*x[23], x[12]*x[24] + x[13]*x[34], 
 
   x[12]*x[25] + x[13]*x[35] + x[14]*x[45]}, 
 
  {0, 0, 0, x[23]*x[34], x[23]*x[35] + x[24]*x[45]}, 
 
  {0, 0, 0, 0, x[34]*x[45]}, {0, 0, 0, 0, 0}, 
 
  {0, 0, 0, 0, 0}}
;[o]
{{0, 0, x[12] x[23], x[12] x[24] + x[13] x[34], 
 
   x[12] x[25] + x[13] x[35] + x[14] x[45]}, 
 
  {0, 0, 0, x[23] x[34], x[23] x[35] + x[24] x[45]}, 
 
  {0, 0, 0, 0, x[34] x[45]}, {0, 0, 0, 0, 0}, 
 
  {0, 0, 0, 0, 0}}
:[font = input; initialization; preserveAspect; startGroup]
*)
T.T.T
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{0, 0, 0, x[12]*x[23]*x[34], 
 
   x[12]*x[23]*x[35] + (x[12]*x[24] + x[13]*x[34])*x[45]}, 
 
  {0, 0, 0, 0, x[23]*x[34]*x[45]}, {0, 0, 0, 0, 0}, 
 
  {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}
;[o]
{{0, 0, 0, x[12] x[23] x[34], 
 
   x[12] x[23] x[35] + (x[12] x[24] + x[13] x[34]) x[45]}, 
 
  {0, 0, 0, 0, x[23] x[34] x[45]}, {0, 0, 0, 0, 0}, 
 
  {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixPower[T,4]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
{{0, 0, 0, 0, x[12]*x[23]*x[34]*x[45]}, {0, 0, 0, 0, 0}, 
 
  {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}
;[o]
{{0, 0, 0, 0, x[12] x[23] x[34] x[45]}, {0, 0, 0, 0, 0}, 
 
  {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}
:[font = input; initialization; preserveAspect]
*)
The powers 2,3,4 of T are not the 0 matrix. Moreover,
they follow the pattern that we observed in our random
matrices.
(*
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
MatrixPower[T,5]//MatrixForm
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, 
 
   {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}]
;[o]
0   0   0   0   0

0   0   0   0   0

0   0   0   0   0

0   0   0   0   0

0   0   0   0   0
:[font = input; initialization; preserveAspect]
*)
Exercise 1.4.13
(*
:[font = input; initialization; preserveAspect]
*)
By induction, we'll show that T^i is stricly upper
triangular and its first i-1 superdiagonals are 0,
if T is strictly upper triangular, which obviously
suffices for our purpose.
i=1 statement follows immediately from T=T^1 strictly
upper triangular.
Suppose T^i strictly upper triangular and having i-1
superdiagonals equal 0. Then the j-th row of T^i will
have (at least) Min[j+i-1,n] leading 0's, whereis the
k-th column of T will always have (at most) k-1 nonzero
entries, from top. That is, in T^(i+1)=(T^i).T, the
[[j,k]]-th entry will be 0 for all j+i >= k, which
proves our statement.
(*
^*)