$Post:=If[MatrixQ[#],MatrixForm[#],#]&

Exercise 3.5.1

A=Table[If[i==j,i+j-1,i+j+1],{i,5},{j,5}]

ES=Eigensystem[A];
lambda=ES[[1]]

Q=ES[[2]]//Together; Q//N

We observe that there are 5 nonzero, linearly independent 
eigenvectors, so A is diagonalizable. (The mention 
"nonzero" from above could be considered redundant, since 
in general we only consider nonzero eigenvectors - the
zero vector being a trivial eigenvector for any eigenvalue
and any square matrix of matching size. However, if the
number of linearly independent eigenvectors does not match
the multiplicity of an eigenvalue, Mathematica inserts a
number of null eigenvectors up to that multiplicity.) 

P=Transpose[Q]; P//TableForm

We will use floating point calculations because of the
radicals which would make the memory fill up.

P=P//N;
T=Inverse[P].A.P; 
T=Table[If[Abs[T[[i,j]]]>0.0000001,T[[i,j]],0],{i,5},{j,5}]

Exercise 3.5.2

A=Table[If[i>j,1,0],{i,4},{j,4}]

ES=Eigensystem[A];
lambda=ES[[1]]

Q=ES[[2]]

Only one nonzero eigenvector ==> A is not diagonalizable.

Exercise 3.5.3

A=Table[If[i==j,2,1],{i,5},{j,5}]

ES=Eigensystem[A];
lambda=ES[[1]]

Q=ES[[2]]

A is diagonalizable.

P=Transpose[Q]

T=Inverse[P].A.P

Exercise 3.5.4

A=Table[i+j-1,{i,5},{j,5}]

ES=Eigensystem[A];
lambda=ES[[1]]

Q=ES[[2]]//Together; Q//N

A is diagonalizable.

P=Transpose[Q]; P//TableForm

P=P//N;
T=Inverse[P].A.P; 
T=Table[If[Abs[T[[i,j]]]>0.0000001,T[[i,j]],0],{i,5},{j,5}]

Exercise 3.5.5

A=Table[i+j,{i,5},{j,5}]

ES=Eigensystem[A];
lambda=ES[[1]]

Q=ES[[2]]//Together; Q//N

A is diagonalizable.

P=Transpose[Q]; P//TableForm

P=P//N;
T=Inverse[P].A.P; 
T=Table[If[Abs[T[[i,j]]]>0.0000001,T[[i,j]],0],{i,5},{j,5}]

Exercise 3.5.6

A=Table[If[i>j,1,0],{i,4},{j,4}]

ES=Eigensystem[A];
ES[[2]]

M=NullSpace[ES[[2]]]

P=Transpose[Join[{ES[[2,1]]},M]]

T=Inverse[P].A.P

Exercise 3.5.7

A=Table[Mod[i+j,2],{i,4},{j,4}]

ES=Eigensystem[A];
ES[[2]]

M=NullSpace[ES[[2]]]

P=Transpose[ES[[2]]]

T=Inverse[P].A.P

Exercise 3.5.8

A=Table[If[i<=2,2,0],{i,4},{j,4}]

ES=Eigensystem[A];
ES[[2]]

M=NullSpace[ES[[2]]]

P=Transpose[ES[[2]]]

T=Inverse[P].A.P

Exercise 3.5.9

A=Table[If[i==j,0,1],{i,4},{j,4}]

ES=Eigensystem[A];
ES[[2]]

M=NullSpace[ES[[2]]]

P=Transpose[ES[[2]]]

T=Inverse[P].A.P