U = 
      [[.6, .28], [.8, -.96]]

inv( U ) = 
      [[1.2, .35], [1, -.75]]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Integer matrices         %
% with integer eigenvalues %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Eigenvalues: 50, -50
U*diag([[50, -50]])*inv( U ) = 
      [[22, 21], [96, -22]]

% Eigenvalues: 49, -51
U*diag([[49, -51]])*inv( U ) = 
      [[21, 21], [96, -23]]

% Eigenvalues: 51, -49
U*diag([[51, -49]])*inv( U ) = 
      [[23, 21], [96, -21]]

% Eigenvalues: 48, -52
U*diag([[48, -52]])*inv( U ) = 
      [[20, 21], [96, -24]]

% Eigenvalues: 52, -48
U*diag([[52, -48]])*inv( U ) = 
      [[24, 21], [96, -20]]

% Eigenvalues: 47, -53
U*diag([[47, -53]])*inv( U ) = 
      [[19, 21], [96, -25]]

% Eigenvalues: 53, -47
U*diag([[53, -47]])*inv( U ) = 
      [[25, 21], [96, -19]]

% Eigenvalues: 46, -54
U*diag([[46, -54]])*inv( U ) = 
      [[18, 21], [96, -26]]

% Eigenvalues: 54, -46
U*diag([[54, -46]])*inv( U ) = 
      [[26, 21], [96, -18]]

% Eigenvalues: 68, -32
U*diag([[68, -32]])*inv( U ) = 
      [[40, 21], [96, -4]]

% Eigenvalues: 69, -31
U*diag([[69, -31]])*inv( U ) = 
      [[41, 21], [96, -3]]

% Eigenvalues: 70, -30
U*diag([[70, -30]])*inv( U ) = 
      [[42, 21], [96, -2]]

% Eigenvalues: 71, -29
U*diag([[71, -29]])*inv( U ) = 
      [[43, 21], [96, -1]]

% Eigenvalues: 72, -28
U*diag([[72, -28]])*inv( U ) = 
      [[44, 21], [96, 0]]

% Eigenvalues: 73, -27
U*diag([[73, -27]])*inv( U ) = 
      [[45, 21], [96, 1]]

% Eigenvalues: 74, -26
U*diag([[74, -26]])*inv( U ) = 
      [[46, 21], [96, 2]]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Matrices with at most 1 digit %
% beyond the decimal point      %
% with integer eigenvalues      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Eigenvalues: 5, -5
U*diag([[50, -50]])*inv( U )/10 = 
      [[2.2, 2.1], [9.6, -2.2]]

% Eigenvalues: 4, -6
U*diag([[40, -60]])*inv( U )/10 = 
      [[1.2, 2.1], [9.6, -3.2]]

% Eigenvalues: 6, -4
U*diag([[60, -40]])*inv( U )/10 = 
      [[3.2, 2.1], [9.6, -1.2]]

% Eigenvalues: 3, -7
U*diag([[30, -70]])*inv( U )/10 = 
      [[.2, 2.1], [9.6, -4.2]]

% Eigenvalues: 7, -3
U*diag([[70, -30]])*inv( U )/10 = 
      [[4.2, 2.1], [9.6, -.2]]

% Eigenvalues: 2, -8
U*diag([[20, -80]])*inv( U )/10 = 
      [[-.8, 2.1], [9.6, -5.2]]

% Eigenvalues: 8, -2
U*diag([[80, -20]])*inv( U )/10 = 
      [[5.2, 2.1], [9.6, .8]]

% Eigenvalues: 1, -9
U*diag([[10, -90]])*inv( U )/10 = 
      [[-1.8, 2.1], [9.6, -6.2]]

% Eigenvalues: 9, -1
U*diag([[90, -10]])*inv( U )/10 = 
      [[6.2, 2.1], [9.6, 1.8]]

% Eigenvalues: 11, 1
U*diag([[110, 10]])*inv( U )/10 = 
      [[8.2, 2.1], [9.6, 3.8]]

% Eigenvalues: 12, 2
U*diag([[120, 20]])*inv( U )/10 = 
      [[9.2, 2.1], [9.6, 4.8]]

% Eigenvalues: 13, 3
U*diag([[130, 30]])*inv( U )/10 = 
      [[10.2, 2.1], [9.6, 5.8]]

% Eigenvalues: 8, -12
U*diag([[80, -120]])*inv( U )/10 = 
      [[2.4, 4.2], [19.2, -6.4]]

% Eigenvalues: 14, 4
U*diag([[140, 40]])*inv( U )/10 = 
      [[11.2, 2.1], [9.6, 6.8]]

% Eigenvalues: 7, -13
U*diag([[70, -130]])*inv( U )/10 = 
      [[1.4, 4.2], [19.2, -7.4]]

% Eigenvalues: 6, -14
U*diag([[60, -140]])*inv( U )/10 = 
      [[.4, 4.2], [19.2, -8.4]]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Matrices with at most 2 digits %
% beyond the decimal point       %
% with integer eigenvalues       %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Eigenvalues: 1, -1
U*diag([[100, -100]])*inv( U )/100 = 
      [[.44, .42], [1.92, -.44]]

% Eigenvalues: 1, -2
U*diag([[100, -200]])*inv( U )/100 = 
      [[.16, .63], [2.88, -1.16]]

% Eigenvalues: 1, 2
U*diag([[100, 200]])*inv( U )/100 = 
      [[1.28, -.21], [-.96, 1.72]]

% Eigenvalues: 2, -1
U*diag([[200, -100]])*inv( U )/100 = 
      [[1.16, .63], [2.88, -.16]]

% Eigenvalues: 2, -2
U*diag([[200, -200]])*inv( U )/100 = 
      [[.88, .84], [3.84, -.88]]

% Eigenvalues: 1, -3
U*diag([[100, -300]])*inv( U )/100 = 
      [[-.12, .84], [3.84, -1.88]]

% Eigenvalues: 1, 3
U*diag([[100, 300]])*inv( U )/100 = 
      [[1.56, -.42], [-1.92, 2.44]]

% Eigenvalues: 3, -1
U*diag([[300, -100]])*inv( U )/100 = 
      [[1.88, .84], [3.84, .12]]

% Eigenvalues: 2, -3
U*diag([[200, -300]])*inv( U )/100 = 
      [[.6, 1.05], [4.8, -1.6]]

% Eigenvalues: 2, 3
U*diag([[200, 300]])*inv( U )/100 = 
      [[2.28, -.21], [-.96, 2.72]]

% Eigenvalues: 3, -2
U*diag([[300, -200]])*inv( U )/100 = 
      [[1.6, 1.05], [4.8, -.6]]

% Eigenvalues: 1, -4
U*diag([[100, -400]])*inv( U )/100 = 
      [[-.4, 1.05], [4.8, -2.6]]

% Eigenvalues: 1, 4
U*diag([[100, 400]])*inv( U )/100 = 
      [[1.84, -.63], [-2.88, 3.16]]

% Eigenvalues: 3, -3
U*diag([[300, -300]])*inv( U )/100 = 
      [[1.32, 1.26], [5.76, -1.32]]

% Eigenvalues: 2, -4
U*diag([[200, -400]])*inv( U )/100 = 
      [[.32, 1.26], [5.76, -2.32]]

% Eigenvalues: 2, 4
U*diag([[200, 400]])*inv( U )/100 = 
      [[2.56, -.42], [-1.92, 3.44]]