The eigen values of a (2 × 2) matrix X are -2 and -3. The eigen values of the matrix (X + $$I$$) (X + 5$$I$$) are A. -3, -4 B. -1, -2 C. -1, -3 D. -2, -4

The correct answer is $\boxed{\text{B. }-1, -2}$.

The eigenvalues of a matrix are the roots of its characteristic polynomial. The characteristic polynomial of a 2×2 matrix $A$ is given by $$p(x) = |xI – A| = x^2 – (tr(A))x + \det(A).$$

In this case, $A$ is a 2×2 matrix with eigenvalues $-2$ and $-3$. Therefore, its characteristic polynomial is $$p(x) = x^2 – (-2 – 3)x + (-2)(-3) = x^2 – 1x + 4.$$

The eigenvalues of the matrix $(X + I)(X + 5I)$ are the roots of the polynomial $$p(x) = (x + 1)(x + 5) = x^2 + 6x + 5.$$

Comparing the two polynomials, we see that the eigenvalues of $(X + I)(X + 5I)$ are $-1$ and $-2$.