Let's go through each part of this question.

Part A: 2x2 Matrix $A$

1. Finding the Maximum and Minimum Determinant for $A$:
   • The determinant of a $2 \times 2$ matrix $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ is given by: $\det(A) = a_{11}a_{22} - a_{12}a_{21}.$
   • Each $a_{ij}$ can range from -10 to 10, so we need to maximize and minimize the product $a_{11}a_{22} - a_{12}a_{21}$.
   • Maximum value: occurs when $a_{11} = 10$, $a_{22} = 10$, $a_{12} = -10$, and $a_{21} = -10$. Thus, $\det(A) = (10)(10) - (-10)(-10) = 100 - 100 = 0.$
   • Minimum value: