(*^
::[	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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:[font = input; initialization; preserveAspect]
*)
$Post:=If[MatrixQ[#],MatrixForm[#],#]&
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.5.1
(*
:[font = input; initialization; preserveAspect]
*)
cofactor[X_,i_,j_]:=(-1)^(i+j)Det[Transpose[Drop[      \
                     Transpose[Drop[X,{i,i}]],{j,j}]]]
(*
:[font = input; initialization; preserveAspect]
*)
adj[X_]:=Transpose[Table[cofactor[X,i,j],{i,Length[X]},\
                                         {j,Length[X]}]]
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Table[i+j,{i,3},{j,3}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{2, 3, 4}, {3, 4, 5}, {4, 5, 6}}]
;[o]
2   3   4

3   4   5

4   5   6
:[font = input; initialization; preserveAspect; startGroup]
*)
lhs=A.adj[A];
rhs=Det[A]IdentityMatrix[3];
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 1.5.2
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[Random[Integer,{-100,100}]&,{5,5}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-47, -45, -84, -6, -41}, 
 
   {-12, -2, -73, -35, 8}, {-9, -20, 83, -21, 43}, 
 
   {-93, 42, -17, -27, 10}, {-58, -48, 10, 80, -79}}]
;[o]
-47   -45   -84   -6    -41

-12   -2    -73   -35   8

-9    -20   83    -21   43

-93   42    -17   -27   10

-58   -48   10    80    -79
:[font = input; initialization; preserveAspect; startGroup]
*)
lhs=A.adj[A];
mhs=adj[A].A;
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=Det[A]IdentityMatrix[5];
mhs-rhs
(*
:[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 1.5.3
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[Random[Integer,{-100,100}]&,{4,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-4, -21, -98, 33}, {-100, 4, -12, -44}, 
 
   {-45, -79, -28, -91}, {-31, -75, -41, 77}}]
;[o]
-4     -21    -98    33

-100   4      -12    -44

-45    -79    -28    -91

-31    -75    -41    77
:[font = input; initialization; preserveAspect; startGroup]
*)
rhs=adj[Transpose[A]];
lhs=Transpose[adj[A]];
lhs-rhs
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{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
:[font = input; initialization; preserveAspect]
*)
Exercise 1.5.4
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[Random[Integer,{-100,100}]&,{8,8}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{43, 98, 11, -39, -39, -88, -41, 43}, 
 
   {48, 20, 94, 14, 38, -76, 97, 2}, 
 
   {-70, 58, 48, -69, 29, 58, 2, -78}, 
 
   {92, -7, -23, 74, -5, -57, -100, 92}, 
 
   {28, 33, -26, -74, -42, -69, 56, -19}, 
 
   {-35, -6, -92, -17, -38, 5, -72, 17}, 
 
   {-35, -5, 0, -28, -54, 30, 70, -26}, 
 
   {-56, -8, 1, 94, -45, 15, 5, -37}}]
;[o]
43     98     11     -39    -39    -88    -41    43

48     20     94     14     38     -76    97     2

-70    58     48     -69    29     58     2      -78

92     -7     -23    74     -5     -57    -100   92

28     33     -26    -74    -42    -69    56     -19

-35    -6     -92    -17    -38    5      -72    17

-35    -5     0      -28    -54    30     70     -26

-56    -8     1      94     -45    15     5      -37
:[font = input; initialization; preserveAspect; startGroup]
*)
B=A[[{2,1,3,4,5,6,7,8}]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{48, 20, 94, 14, 38, -76, 97, 2}, 
 
   {43, 98, 11, -39, -39, -88, -41, 43}, 
 
   {-70, 58, 48, -69, 29, 58, 2, -78}, 
 
   {92, -7, -23, 74, -5, -57, -100, 92}, 
 
   {28, 33, -26, -74, -42, -69, 56, -19}, 
 
   {-35, -6, -92, -17, -38, 5, -72, 17}, 
 
   {-35, -5, 0, -28, -54, 30, 70, -26}, 
 
   {-56, -8, 1, 94, -45, 15, 5, -37}}]
;[o]
48     20     94     14     38     -76    97     2

43     98     11     -39    -39    -88    -41    43

-70    58     48     -69    29     58     2      -78

92     -7     -23    74     -5     -57    -100   92

28     33     -26    -74    -42    -69    56     -19

-35    -6     -92    -17    -38    5      -72    17

-35    -5     0      -28    -54    30     70     -26

-56    -8     1      94     -45    15     5      -37
:[font = input; initialization; preserveAspect; startGroup]
*)
lhs=Det[B];
rhs=-Det[A];
lhs-rhs
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect]
*)
Exercise 1.5.5
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[Random[Integer,{-100,100}]&,{6,6}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{44, 86, 47, -21, 1, -9}, 
 
   {-5, -62, -43, -7, -13, 58}, 
 
   {-15, -59, 98, -11, -22, 32}, 
 
   {-23, 51, 31, -94, 28, -18}, {-9, -4, 67, -69, 90, -90}, 
 
   {0, 22, -18, 50, -48, 55}}]
;[o]
44    86    47    -21   1     -9

-5    -62   -43   -7    -13   58

-15   -59   98    -11   -22   32

-23   51    31    -94   28    -18

-9    -4    67    -69   90    -90

0     22    -18   50    -48   55
:[font = input; initialization; preserveAspect; startGroup]
*)
lhs=Det[A];
A[[2]]=A[[2]]+k*A[[1]];
rhs=Det[A];
lhs-rhs
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect]
*)
Exercise 1.5.6
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[Random[Integer,{-100,100}]&,{7,7}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-58, -18, 91, 61, -61, 90, 7}, 
 
   {-27, -97, 82, 52, -42, -50, 47}, 
 
   {-69, -53, 60, 22, -3, 63, 94}, 
 
   {71, 31, 17, 68, 9, -32, 11}, 
 
   {-29, -11, -38, 40, -56, 40, 15}, 
 
   {15, -39, -74, -93, 52, 75, 54}, 
 
   {42, 54, 13, 61, 22, 65, 48}}]
;[o]
-58   -18   91    61    -61   90    7

-27   -97   82    52    -42   -50   47

-69   -53   60    22    -3    63    94

71    31    17    68    9     -32   11

-29   -11   -38   40    -56   40    15

15    -39   -74   -93   52    75    54

42    54    13    61    22    65    48
:[font = input; initialization; preserveAspect; startGroup]
*)
Det[A]-Det[Transpose[A]]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect]
*)
Exercise 1.5.7
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[Random[Integer,{-100,100}]&,{4,4}]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{26, 60, 36, -78}, {48, -76, -98, 51}, 
 
   {-68, -75, -62, 7}, {73, -39, 6, -90}}]
;[o]
26    60    36    -78

48    -76   -98   51

-68   -75   -62   7

73    -39   6     -90
:[font = input; initialization; preserveAspect]
*)
a)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
lhs=Det[A];
rhs:=Sum[A[[i,j]]*cofactor[A,i,j],{j,1,4}];
Do[Print[lhs-rhs],{i,1,4}]
(*
:[font = print; inactive; preserveAspect; endGroup]
0
0
0
0
:[font = input; initialization; preserveAspect]
*)
b)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
lhs=Det[A];
rhs:=Sum[A[[i,j]]*cofactor[A,i,j],{i,1,4}]
Do[Print[lhs-rhs],{j,1,4}]
(*
:[font = print; inactive; preserveAspect; endGroup]
0
0
0
0
:[font = input; initialization; preserveAspect]
*)
c)
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
lhs:=Sum[A[[r,j]]*cofactor[A,s,j],{j,1,4}]
rhs:=Sum[A[[i,r]]*cofactor[A,i,s],{i,1,4}]
Do[Do[Print[lhs-rhs," for r-s=",r-s],{s,1,4}],{r,1,4}]
(*
:[font = print; inactive; preserveAspect; endGroup]
0 for r-s=0
0 for r-s=-1
0 for r-s=-2
0 for r-s=-3
0 for r-s=1
0 for r-s=0
0 for r-s=-1
0 for r-s=-2
0 for r-s=2
0 for r-s=1
0 for r-s=0
0 for r-s=-1
0 for r-s=3
0 for r-s=2
0 for r-s=1
0 for r-s=0
:[font = input; initialization; preserveAspect]
*)
(Note that the equality holds even for r=s)
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.5.8
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[a,{4,4}];
lhs=Det[A];
rhs=Det[Transpose[A]];
lhs-rhs
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect]
*)
Exercise 1.5.9
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
rhs=adj[Transpose[A]];
lhs=Transpose[adj[A]];
lhs-rhs
(*
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{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
:[font = input; initialization; preserveAspect]
*)
Exercise 1.5.10
(*
:[font = input; initialization; preserveAspect]
*)
We'll use induction over the dimension n for both
identities in parallel.
For n=1 they are both obvious, since the transpose of
a 1*1 matrix is that matrix itself.
Suppose now both identities are true for dimension n.
Then, we look at an arbitrary (n+1)*(n+1) matrix A. 
Since we supposed Det[B]=Det[Transpose[B]] for any n*n 
matrix B, we hence have that the [[i,j]] cofactor of A 
is the [[j,i]] cofactor of Transpose[A] (note that those
cofactors are computed from transposed n*n matrices).
So, for any (n+1)*(n+1) matrix A we have:
Transpose[Adj[A]]=Adj[Transpose[A]].
To the determinants: Since the textbook gives us the
identity: A.Adj[A]=Det[A]*IdentityMatrix, stated in
ex. 2, we'll use it in this proof, rather than some
definition of Det not present in the textbook.
Denote Id the IdentityMatrix and T[A] the transpose 
of A.
Det[A]*Id=T[Det[A]*Id]=T[Adj[A].A]=T[A].T[Adj[A]]=
         =T[A].Adj[T[A]]=Det[T[A]]*Id
(*
:[font = input; initialization; preserveAspect]
*)
Exercise 1.5.11
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
lhs=Det[A];
B=A[[{2,1,3,4}]];
rhs=-Det[B];
lhs-rhs
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect]
*)
Exercise 1.5.12
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[a,{4,4}];
lhs=Det[A];
A[[2]]=A[[2]]+k*A[[1]];
rhs=Det[A];
lhs-rhs
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
:[font = input; initialization; preserveAspect]
*)
Exercise 1.5.13
(*
:[font = input; initialization; preserveAspect; startGroup]
*)
A=Array[a,{4,4}];
lhs=k*Det[A];
A[[1]]=k*A[[1]];
rhs=Det[A];
Simplify[lhs-rhs]
(*
:[font = output; output; inactive; preserveAspect; endGroup]
0
;[o]
0
^*)