Positive diagonal entries in $2 \times 2$ matrix

Question

I am facing the following problem:

A symmetric $2 \times 2$ matrix $A$ has eigenvalues $\lambda_1 = 3$ and $\lambda_2 = 4$. Compute the determinant and trace of $A$. Is the following statement true, false or depends on the particular entries of $A$?

"All diagonal entries of $A$ are positive."

I know that $\det(A) = 12$ and $\mbox{tr}(A) = 7$. How can I determine the signs of the diagonal entries?

Answer 1

Let the matrix be $\begin{pmatrix}a&b \\b&d\\\end{pmatrix}$.

Then $a+d=7$ and $ad-b^2=12$. Therefore $a$ and $d$ are both positive.