To do this, we'll solve the characteristic equation:
$$\text{det}(A - \lambda I) = 0$$
For the given matrix A, this equation becomes:
$$\text{det}\begin{pmatrix} 1-\lambda & 2 \\ 4 & 3-\lambda \end{pmatrix} = (1-\lambda)(3-\lambda) - (2)(4) = 0$$
Simplifying,
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