M is a 2 × 2 matrix with eigen values 4 and 9. The eigen values of M2 are A. -2 and -3 B. 2 and 3 C. 4 and 9 D. 16 and 81

The correct answer is $\boxed{\text{D. }16 \text{ and }81}$.

The eigenvalues of a matrix are the roots of its characteristic polynomial. The characteristic polynomial of a 2×2 matrix $M$ is given by:

$$p(x) = |xI – M|$$

where $I$ is the identity matrix.

If the eigenvalues of $M$ are $\lambda_1$ and $\lambda_2$, then the eigenvalues of $M^2$ are $\lambda_1^2$ and $\lambda_2^2$.

In this case, the eigenvalues of $M$ are $4$ and $9$, so the eigenvalues of $M^2$ are $4^2 = 16$ and $9^2 = 81$.