Consider the following $2 \times 2$ matrix $A$ where two elements are unknown and are marked by $a$ and $b$. The eigenvalues of this matrix are -1 and 7. What are the values of $a$ and $b$? $$A = \begin{pmatrix} 1 & 4 \\ b & a \end{pmatrix}$$

Let the given matrix be:

The eigenvalues are given as $\lambda_1 = -1$ and $\lambda_2 = 7$.

1. Sum of Eigenvalues equals the Trace of the matrix:

   $$\operatorname{Trace}(A) = 1 + a$$

   $$\lambda_1 + \lambda_2 = -1 + 7 = 6$$

   $$1 + a = 6 \implies a = 5$$

2. Product of Eigenvalues equals the Determinant of the matrix:

   $$\det(A) = 1 \cdot a - 4 \cdot b = a - 4b$$

   $$\lambda_1 \cdot \lambda_2 = (-1) \cdot 7 = -7$$

   Substitute $a = 5$:

   $$5 - 4b = -7 \implies 4b = 12 \implies b = 3$$

Thus, $a = 5$ and $b = 3$, which corresponds to option D [21].