Question

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

Answer 1

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

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

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

$\lambda^2-\mathrm{tr}(\mathbf A)\lambda+\det(\mathbf A)$

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

$\lambda^2-15\lambda+50$

and the eigenvalues are those $\lambda$ for which the characteristic polynomial evaluates to 0.

$\lambda^2-15\lambda+50=(\lambda-5)(\lambda-10)=0\implies\lambda=5,\lambda=10$