We are tasked with calculating $2A - B$ for the given matrices:

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

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 $2A$

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

Perform the subtraction element-wise: $2A - B = \begin{pmatrix} 4 - (-3) & 8 - (-9) \\ 6 - (-2) & 18 - (-4) \end{pmatrix} = \begin{pmatrix} 4 + 3 & 8 + 9 \\ 6 + 2 & 18 + 4 \end{pmatrix}.$

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

Final Answer:

$\boxed{\begin{pmatrix} 7 & 17 \\ 8 & 22 \end{pmatrix}}.$

Let me know if you'd like more detailed explanations or clarifications!

Here are 5 related questions to explore:

1. What is the determinant of the matrix $2A - B$?
2. How do you calculate the transpose of $2A - B$?
3. Can $2A - B$ be diagonalized? If yes, how?
4. What are the eigenvalues of $2A - B$?
5. How is matrix addition and subtraction applied in practical contexts?

Tip: Always verify matrix dimensions before performing operations to ensure they are compatible!