Alternative way to find the eigenvalues of a matrix.

Question

I am trying questions of Masters of Mathematics Entrance exam of my university and I am looking for an alternative solution for this question.

Find Eigenvalue of the Matrix $$ \begin {bmatrix} 1 & 1 & 2 \\ 1 & -2 & -5 \\ 2 & 5 & -3 \\ \end {bmatrix} $$ Options

• $-4, 3,-3$
• $4,3,1$
• $4,-4+\sqrt{13} , -4-\sqrt{13}$
• $4,-2+\sqrt{7} , -2-\sqrt{7}$

I know that one way to find eigenvalues is using Cayley-Hamilton Theorem and then equating characteristic polynomial to $O$ .

But since Exam is Objective and Not Subjective, Is there any way to find Eigenvalues without using Cayley-Hamilton theorem (as that method is Time-Consuming)?

If yes kindly shed some light.

Edit: All Four options given are same as options given in assignment. So, maybe question is wrong.

Answer 1

Hints: try to find trace and det of the matrix $A$ , that will eliminate all other false options. [Rule : trace of $A$ = sum of all eigenvalues of $A$, det($A$)=multiplication of eigenvalues of $A$]