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This question was asked in my Linear Algebra assignment and I was unable to solve it.

Let $a, b, c \in \Bbb R_{>0}$ such that $b^{2} + c^{2} < a < 1$. Consider the following $3 \times 3$ matrix $$A = \begin {bmatrix} 1&b&c\\b&a&0\\c&0&1\end{bmatrix}$$ Prove that the eigenvalues of $A$ cannot be non-real complex numbers.

I tried by using formula of trace= sum of eigenvalues and that product of eigenvalues = determinant but trace formula but that does not seem to help. Can anyone please tell how should i proceed.

Rodrigo de Azevedo
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    Don't all real symmetric matrices have real eigenvalues? – 5Pack Aug 08 '20 at 13:32
  • @5Pack thank you !!!! –  Aug 08 '20 at 13:36
  • Related: https://math.stackexchange.com/q/354115 – Rodrigo de Azevedo Aug 08 '20 at 13:39
  • @RodrigodeAzevedo If you have some spare time can you please have a look at this question asked by me . I shall be really thankful to you .https://math.stackexchange.com/questions/3788633/a-quiz-question-based-on-matrices-on-complex-field –  Aug 17 '20 at 17:14

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