GATE EC 2022 Set 1 Q27 — Linear Algebra

Let $\alpha, \beta$ be two non-zero real numbers and $v_{1}, v_{2}$ be two non-zero real vectors of size $3 \times 1.$ Suppose that $v_{1}$ and $v_{2}$ satisfy $v_{1}^{T} v_{2} = 0, v_{1}^{T} v_{1} = 1,$ and $v_{2}^{T} v_{2} = 1.$ Let $A$ be the $3 \times 3$ matrix given by:

$$A = \alpha v_{1} v_{1}^{T} + \beta v_{2} v_{2}^{T}$$

The eigenvalues of $A$ are ________________.