Exercise  1 Section 6.9 in G&S
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Solve[Det[{{-mu,mu},{lambda,-lambda}}-x{{1,0},{0,1}}]==0,{x}]

Solve[{{-mu,mu},{lambda,-lambda}}.{{b11,b12},{b21,b22}}=={{b11,b12},{b21,b22}}.{{0,0},{0,-lambda-mu}},{b11,b12,b21,b22}]

B={{1,mu},{1,-lambda}};

FullSimplify[B.{{1,0},{0,Exp[-t*(lambda+mu)]}}.Inverse[B]]