To solve the problem $2A - B$, we first calculate $2A$ and then subtract $B$ from it.

Given: $A = \begin{pmatrix} 2 & 4 \\ 3 & 9 \end{pmatrix}, \quad B = \begin{pmatrix} -3 & -9 \\ -2 & -4 \end{pmatrix}$

Step 1: Calculate $2A$

$2A = 2 \cdot \begin{pmatrix} 2 & 4 \\ 3 & 9 \end{pmatrix} = \begin{pmatrix} 4 & 8 \\ 6 & 18 \end{pmatrix}$

Step 2: Subtract $B$ from $2A$

$2A - B = \begin{pmatrix} 4 & 8 \\ 6 & 18 \end{pmatrix} - \begin{pmatrix} -3 & -9 \\ -2 & -4 \end{pmatrix}$

Subtract element-wise: $2A - B = \begin{pmatrix} 4 - (-3) & 8 - (-9) \\ 6 - (-2) & 18 - (-4) \end{pmatrix} = \begin{pmatrix} 7 & 17 \\ 8 & 22 \end{pmatrix}$

Final Answer:

$2A - B = \begin{pmatrix} 7 & 17 \\ 8 & 22 \end{pmatrix}$

Would you like further explanations or any additional calculations?

Here are 5 related questions to explore further:

1. What is the determinant of the resulting matrix $2A - B$?
2. How would the result change if $A$ were a 3x3 matrix?
3. What would happen if you calculated $A^2 - B^2$ instead?
4. How can matrix subtraction be represented geometrically?
5. What are the eigenvalues of $2A - B$, if any?

Tip: When working with matrices, ensure all operations (addition, subtraction, scalar multiplication) are element-wise unless specified otherwise!