The trace and determinant of a 2 × 2 matrix are known to be -2 and -35 respectively. It eigen values are A. -30 and -5 B. -37 and -1 C. -7 and 5 D. 17.5 and -2

The correct answer is $\boxed{\text{A}. -30, -5}$.

The eigenvalues of a 2×2 matrix can be found using the following formula:

$$\lambda_1, \lambda_2 = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$

where $a$ is the trace of the matrix, $b$ is the determinant of the matrix, and $c$ is the product of the eigenvalues.

In this case, we know that $a = -2$ and $b = -35$. Substituting these values into the formula, we get:

$$\lambda_1, \lambda_2 = \frac{-(-2) \pm \sqrt{(-(-2))^2 – 4(-35)}}{2(-2)}$$

$$\lambda_1, \lambda_2 = \frac{2 \pm \sqrt{144}}{-4}$$

$$\lambda_1, \lambda_2 = \frac{2 \pm 12}{-4}$$

$$\lambda_1, \lambda_2 = -30, -5$$

Therefore, the eigenvalues of the matrix are $-30$ and $-5$.

The other options are incorrect because they do not correspond to the eigenvalues of the matrix.