linear algebra - Find eigenvector and eigenvelue of 2 by 2 symmetric matrix, given another eigenvector and eigenvalue

A symmetric 2 by 2 matrix has eigenvalue -4 and eigenvector(-2;3). Find another eigenvector and eigenvalue and construct orthonormal basis(of eigenvectors of matrix A).

Is it possible to find eigenvector not knowing the Matrix(only size and symmetry is given)?

When it is possible, what should i do?
Many thanks

Answer

Hint: The spectral theorem tells us that every symmetric matrix has an orthonormal basis of eigenvectors. This fact allows us to use one eigenvector to find another.

From this question, however, we don't have enough information to figure out what the other eigenvalue is.

In fact, we can deduce that every matrix satisfying the conditions for this question will have the form

$$-\frac{4}{13}\pmatrix{4&-6\\-6&9} +\frac{\lambda}{13} \pmatrix{9&6\\6&4}$$
Where $\lambda$ is the second eigenvalue.