3437.sagews

A = Matrix([[16,-120,240,-140],[-120,1200,-2700,1680],[240,-2700,6480,-4200],[-140,1680,-4200,2800]])

Lambda=A.eigenvalues();print Lambda

[0.666571444687562?, 5.912219379112548?, 148.4059654605107?, 10341.01524371569?]

A.determinant()

6048000

prod(Lambda)

6.048000000000000?e6

L = matrix(SR, 4, 4, var('z'))

A-L

[ -z + 16 -120 240 -140] [ -120 -z + 1200 -2700 1680] [ 240 -2700 -z + 6480 -4200] [ -140 1680 -4200 -z + 2800]

(A-L).determinant()

z^4 - 10496*z^3 + 1603680*z^2 - 10137600*z + 6048000

eqn = (A-L).determinant() == 0

soln = solve(eqn,z, solution_dict=True)

rsoln = [real(s[z]) for s in soln]

print E

[0.666571444687562?, 5.912219379112548?, 148.4059654605107?, 10341.01524371569?]

print [r.n(53) for r in rsoln]

[0.666571446617127, 5.91221937718274, 148.405965460510, 10341.0152437157]

print [r.n(80) for r in rsoln]

[0.66657144468756184504637, 5.9122193791125476547108, 148.40596546051063267796, 10341.015243715689257822]

print [r.n(120) for r in rsoln]

[0.66657144468756184999257851409765940, 5.9122193791125476497665091957118484, 148.40596546051063267795191643369852, 10341.015243715689257822288995856492]

print [r.n(120)-r.n(80) for r in rsoln]

[4.9462133267251470497994e-18, -4.9443033666907621088982e-18, -4.8704394467122269496429e-21, 1.3552527156068805425093e-20]

print [r.n(120)-r.n(53) for r in rsoln]

[-1.92956484124096e-9, 1.92981097768552e-9, 7.95807864051312e-13, -1.81898940354586e-12]