Let's solve the problem.

We are given matrices $A$ and $B$:

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

We need to compute $2A - B$.

Step 1: Compute $2A$

Multiply every element in $A$ by 2: $2A = 2 \cdot \begin{pmatrix} 2 & 4 \\ 3 & 9 \end{pmatrix} = \begin{pmatrix} 4 & 8 \\ 6 & 18 \end{pmatrix}.$

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

Now subtract $B$ element-wise from $2A$: $2A - B = \begin{pmatrix} 4 & 8 \\ 6 & 18 \end{pmatrix} - \begin{pmatrix} -3 & 9 \\ -2 & -4 \end{pmatrix}.$

Perform the subtraction: $2A - B = \begin{pmatrix} 4 - (-3) & 8 - 9 \\ 6 - (-2) & 18 - (-4) \end{pmatrix} = \begin{pmatrix} 7 & -1 \\ 8 & 22 \end{pmatrix}.$

Final Answer:

$2A - B = \begin{pmatrix} 7 & -1 \\ 8 & 22 \end{pmatrix}.$

This corresponds to option (3).

Let me know if you need further details!

Questions to Explore:

1. How do matrix addition and subtraction work element-wise?
2. How does scalar multiplication affect the elements of a matrix?
3. What are the properties of matrix operations (associativity, commutativity)?
4. Can you generalize $nA + B$ for $n \in \mathbb{R}$?
5. Why does the order of operations matter in matrix computations?

Tip: Always double-check arithmetic when dealing with matrix operations to avoid small calculation errors.