How can I find the eigenvalues of this 3x3 matrix

Question

I have a test in Algebra in a few days and a problem with eigenvalues will be there. I tried many ways to solve it and none helped me. How could you find the eigenvalues of the following matrix?

$$A=\begin{pmatrix}8 &-6 &2\\-6 &7 &-4\\2 &-4 &3\end{pmatrix}$$

I've tried add/substract rows and columns and the way this pdf shows https://www.scss.tcd.ie/~dahyotr/CS1BA1/SolutionEigen.pdf

Answer 1

I suppose you have some (course) notes on the topic? You find the eigenvalues by solving: $$\det\left(A-\lambda I_n\right)=0$$ where $A$ is the $n\times n$-matrix given and $I_n$ is the identity matrix. In your case, this means solving: $$\begin{vmatrix} 8-\lambda & -6 & 2 \\ -6& 7-\lambda & -4 \\ 2 & -4 & 3-\lambda \end{vmatrix}=0$$

I've tried add/substract rows and columns and the way this pdf shows

Do you know how to calculate a ($3 \times 3$) determinant?

You can use properties of determinants to try and simplify the calculations, but you can also go for the straightforward calculation by expanding along a column or row of your choice: that will lead to a 3rd degree polynomial equation in $\lambda$.