Suppose that the trace of a 2×2 matrix a is tr(a)=15 and the determinant is det(a)=50. find the eigenvalues ofa.

Recall that the characteristic polynomial of a 2x2 matrix [tex]\mathbf A=\begin{bmatrix}a&b\\c&d\end{bmatrix}[/tex] is

[tex]\det(\mathbf A-\lambda\mathbf I)=\begin{vmatrix}a-\lambda&b\\c&d-\lambda\end{vmatrix}=(a-\lambda)(d-\lambda)-bc=\lambda^2-(a+d)\lambda+(ad-bc)[/tex]

but [tex]\det(\mathbf A)=ad-bc[/tex] and [tex]\mathrm{tr}(\mathbf A)=a+d[/tex], so the characteristic polynomial for [tex]\mathbf A[/tex] is

[tex]\lambda^2-\mathrm{tr}(\mathbf A)\lambda+\det(\mathbf A)[/tex]

We're given that the trace is 15 and determinant is 50, so the characteristic polynomial for the matrix in question is

[tex]\lambda^2-15\lambda+50[/tex]

and the eigenvalues are those [tex]\lambda[/tex] for which the characteristic polynomial evaluates to 0.

[tex]\lambda^2-15\lambda+50=(\lambda-5)(\lambda-10)=0\implies\lambda=5,\lambda=10[/tex]