Necessary and sufficient condition for a matrix to have postive eigenvalues

Question

I have a question that requires the condition for $k$ for the below matrix to have positive eigenvalues.

$\begin{bmatrix} 2 & 1\\ 1 & k\\ \end{bmatrix}$

How to do it ?

Edit: I need a generalized answer for any nxn matrix not just 2x2.

Answer 1

You can use Sylvester's criterion. It works for symmetric matrices of any dimension.

Answer 2

To compute the eigenvalues for a matrix $M$ you can find the determinant of the difference between $M$ and the matrix with $\lambda$ on the diagonal, that is $\lambda I$. Then set this equation to $0$ and solve for the values of $\lambda$.
Like the following:
$$det(M-\lambda I)=0$$

From this you should get an equation in $k$, and I think you can take it from there.