The minimum eigenvalue of the matrix $A$ is $\lambda_1 = -2.5$.
To find the eigenvalues of a matrix, we can use the following formula:
$$\lambda_i = \frac{1}{|A – \lambda_i I|}$$
where $I$ is the identity matrix and $|A|$ is the determinant of $A$.
In this case, we have:
$$|A – \lambda_i I| = \begin{vmatrix} 3 – \lambda_i & 5 & 2 \ 5 & 12 – \lambda_i & 7 \ 2 & 7 & 5 – \lambda_i \end{vmatrix}$$
Expanding the determinant, we get:
$$|A – \lambda_i I| = -\lambda_i^3 + 30 \lambda_i^2 – 257 \lambda_i + 1260$$
We can then use the quadratic formula to solve for the eigenvalues:
$$\lambda_i = \frac{-30 \pm \sqrt{30^2 – 4 \cdot (-1) \cdot 1260}}{2 \cdot (-1)}$$
$$\lambda_i = \frac{-30 \pm \sqrt{12960}}{-2}$$
$$\lambda_i = \frac{-30 \pm 113}{-2}$$
$$\lambda_i = -2.5 \pm 56.5$$
Therefore, the minimum eigenvalue of $A$ is $\lambda_1 = -2.5$.