Eigen Values that aren't expected

Question

$2\times 2$ matrix has the vectors $\begin{bmatrix}2 & \sqrt2 \\ \sqrt2 & 2 \end{bmatrix}$. I'm told the eigen values are $0$ and $3$ from Ron Larson linear algebra $8$ ed book.

How do you get that? I get $\pm \sqrt2 +2$.

Answer 1

You were given wrong information, an easy test is to check the trace which is equal to the sum of the eigenvalues.

The trace of the matrix is $2+2=4$ but the sum of the claimed eigenvalues is $0+3=3$. Hence, the information can't be correct.

Now, to check your answer, $(\sqrt2+2)+(-\sqrt2+2)=4$, so your proposed eigenvalues fit the trace condition.

Now, let's check the determinant of the matrix which is equal to $2^2-\sqrt2^2=2$ and the product of the proposed eigenvalues are $(2+\sqrt2)(2-\sqrt2)=4-2=2$.

Your solution is correct.