# Calculate eigenvalues as a function

I don't understand these two questions, someone can explain how to solve them? Thanks in advance.

in problem 2, I tried to solve it by this way:

## 1 Answer

The eigenvalues are the complex numbers $\lambda$ such that $$\det(A-\lambda I) = 0.$$ We see $$\det(A - \lambda I ) =\left\lvert \begin{matrix} 1-\lambda & q \\ 4 & 2-\lambda \end{matrix} \right\rvert = (\lambda - 2)(\lambda - 1) -4q = \lambda^2 - 3\lambda +(2- 4q).$$ Solving for the roots gives $$\lambda = \tfrac 1 2 \left( 3 \pm \sqrt{9 - 4(2 - 4q)} \right) = \tfrac 1 2 \left( 3 \pm \sqrt{1 + 16q} \right).$$ This answers (1). To answer (2), we cna just set $\lambda = 5$ and solve for $q$. If $\lambda = 5$, then we need $$10 = 3 + \sqrt{1+16q} \implies 7 = \sqrt{1+16q} \implies 49 = 1 + 16q \implies 16q = 48 \implies q =3.$$

• Thank you very much, it makes sense :)
– AdiT
Mar 14, 2016 at 22:46