The problem asks to find matrix $G$ based on two given conditions. Let's analyze and solve the two parts one by one.

Part (a):

The equation given is:

$\begin{pmatrix} 4 & 1 & 2 \\ -1 & 3 & 0 \end{pmatrix} + 2G = \begin{pmatrix} -2 & 1 & 4 \\ 9 & 1 & 6 \end{pmatrix}$

Steps:

1. Isolate $G$: $2G = \begin{pmatrix} -2 & 1 & 4 \\ 9 & 1 & 6 \end{pmatrix} - \begin{pmatrix} 4 & 1 & 2 \\ -1 & 3 & 0 \end{pmatrix}$

   Calculate the matrix subtraction: $\begin{pmatrix} -2 & 1 & 4 \\ 9 & 1 & 6 \end{pmatrix} - \begin{pmatrix} 4 & 1 & 2 \\ -1 & 3 & 0 \end{pmatrix} = \begin{pmatrix} -6 & 0 & 2 \\ 10 & -2 & 6 \end{pmatrix}$

2. Divide by 2 to find $G$: $G = \frac{1}{2} \begin{pmatrix} -6 & 0 & 2 \\ 10 & -2 & 6 \end{pmatrix} = \begin{pmatrix} -3 & 0 & 1 \\ 5 & -1 & 3 \end{pmatrix}$

Thus, the solution for part (a) is: $G = \begin{pmatrix} -3 & 0 & 1 \\ 5 & -1 & 3 \end{pmatrix}$

Part (b):

The equation given is:

$3G - 2\begin{pmatrix} 1 & 1 & 0 \\ -2 & 2 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 6 & 5 \\ -2 & -1 & 0 \end{pmatrix}$

Steps:

1. Expand the equation: $3G - 2\begin{pmatrix} 1 & 1 & 0 \\ -2 & 2 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 6 & 5 \\ -2 & -1 & 0 \end{pmatrix}$

   Calculate the multiplication: $2\begin{pmatrix} 1 & 1 & 0 \\ -2 & 2 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 2 & 0 \\ -4 & 4 & 0 \end{pmatrix}$

   Thus, the equation becomes: $3G - \begin{pmatrix} 2 & 2 & 0 \\ -4 & 4 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 6 & 5 \\ -2 & -1 & 0 \end{pmatrix}$

2. Isolate $3G$: $3G = \begin{pmatrix} 1 & 6 & 5 \\ -2 & -1 & 0 \end{pmatrix} + \begin{pmatrix} 2 & 2 & 0 \\ -4 & 4 & 0 \end{pmatrix}$

   Add the matrices: $3G = \begin{pmatrix} 3 & 8 & 5 \\ -6 & 3 & 0 \end{pmatrix}$

3. Divide by 3 to find $G$: $G = \frac{1}{3} \begin{pmatrix} 3 & 8 & 5 \\ -6 & 3 & 0 \end{pmatrix} = \begin{pmatrix} 1 & \frac{8}{3} & \frac{5}{3} \\ -2 & 1 & 0 \end{pmatrix}$

Thus, the solution for part (b) is: $G = \begin{pmatrix} 1 & \frac{8}{3} & \frac{5}{3} \\ -2 & 1 & 0 \end{pmatrix}$

Would you like further details on any steps or have questions about matrix operations?

Here are 5 follow-up questions you could explore:

1. What are some properties of matrix addition and subtraction?
2. How does matrix scalar multiplication work?
3. Can we find the determinant of matrix $G$ in part (a)?
4. How do we verify the solutions by substituting $G$ back into the original equations?
5. What is the significance of the identity matrix in matrix equations?

Tip: When working with matrices, always ensure that the dimensions match for addition, subtraction, or multiplication operations.