To do this, we need to solve the characteristic equation, which is given by the determinant of (A - λI), where λ is the eigenvalue and I is the identity matrix.
A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}
Now, we find (A - λI):
(A - λI) = \begin{bmatrix} 1
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