A = {{2,5,-1},{5,-3,8},{-1,8,-7}};

N[Eigenvalues[A],10]

N[Solve[Det[A-λ*IdentityMatrix[3]]==0,λ],15]

N[Solve[Det[A-λ*IdentityMatrix[3]]==0,λ],20]

N[Solve[Det[A-λ*IdentityMatrix[3]]==0,λ],30]

Chop[N[Solve[Det[A-λ*IdentityMatrix[3]]==0,λ],15]]

N[Eigenvalues[A],10]

N[Eigenvectors[A],10]

(Dmat = IdentityMatrix[3]*N[Eigenvalues[A],10])//MatrixForm

(Q = Transpose[N[Eigenvectors[A],10]])//MatrixForm

Q.Dmat.Inverse[Q]//MatrixForm

A//MatrixForm

(eDmat = DiagonalMatrix[Exp[N[Eigenvalues[A],10]*t]])//MatrixForm

x0 = {{-12},{4},{7}};

(XSoln = Simplify[Q.eDmat.Inverse[Q].x0])//MatrixForm

XSoln/.t->0

Simplify[D[XSoln,t]-A.XSoln]

ParametricPlot3D[Flatten[XSoln],{t,-0.15,0.15}]

Flatten[XSoln]

(F = {{Cos[t]},{-Sin[t]},{Exp[3t]}})//MatrixForm

(eMDmat = DiagonalMatrix[Exp[-N[Eigenvalues[A],10]*t]])//MatrixForm

eDmat//MatrixForm

(Z = Integrate[Q.eMDmat.Inverse[Q].F,t])//MatrixForm

Cval = Chop[Simplify[Inverse[Q].x0-Z/.t->0]]

(NewXSoln = Chop[Simplify[Q.eDmat.Inverse[Q].(Z+Cval)],10^(-8)])//MatrixForm

Chop[Simplify[N[D[NewXSoln,t]-(A.NewXSoln+F)]]]

NewXSoln/.t->0