Exercise 3.1.1

B=Array[Min,{2,2}]; B//MatrixForm

x*IdentityMatrix[2]-B//MatrixForm

p=Det[%]

lambda=x/.Solve[p==0,x]

NullSpace[lambda[[1]]IdentityMatrix[2]-B]

NullSpace[lambda[[2]]IdentityMatrix[2]-B]

Exercise 3.1.2

A=Array[1&,{9,9}]; A//MatrixForm

lambda=x/.Solve[Det[x*IdentityMatrix[9]-A]==0,x]

NullSpace[lambda[[1]]IdentityMatrix[9]-A]

NullSpace[lambda[[2]]IdentityMatrix[9]-A]

Exercise 3.1.3

Q=Table[i/j,{i,3},{j,3}]; Q//MatrixForm

lambda=x/.Solve[Det[x*IdentityMatrix[3]-Q]==0,x]

All eigenvalues are obviously real.

NullSpace[lambda[[1]]IdentityMatrix[3]-Q]

NullSpace[lambda[[2]]IdentityMatrix[3]-Q]

<<LinearAlgebra`Master`
$Post:=If[MatrixQ[#],MatrixForm[#],#]&

Exercise 3.1.4

M[n_]:=Table[If[i==j+1,1,0],{i,n},{j,n-1}]

a)

A=AppendRows[M[3],Transpose[{{0,-1,-1}}]]

Det[x*IdentityMatrix[3]-A]

b)

A=AppendRows[M[2],{{-1},{2}}]

Det[x*IdentityMatrix[2]-A]

c)

A=AppendRows[M[4],Transpose[{{-3,0,4,0}}]]

Det[x*IdentityMatrix[4]-A]

d)

A=AppendRows[M[3],Transpose[{{1,-3,3}}]]

Det[x*IdentityMatrix[3]-A]