https://file.aiquickdraw.com/mathbot/file/17320432716384hbd4q9y.jpg

To calculate \( 7A - 2B \), we need to multiply matrix \( A \) by 7 and matrix \( B \) by 2, then subtract the resulting matrices.

Here are the matrices:
\[
A = \begin{bmatrix} 2 & 4 \\ 8 & 3 \\ 1 & 9 \\ 0 & 3 \end{bmatrix}, \quad
B = \begin{bmatrix} 8 & 4 \\ 1 & 2 \\ -1 & 3 \\ -4 & 1 \end{bmatrix}.
\]

### Step-by-Step Calculation:
1. Compute \( 7A \):
   \[
   7A = 7 \cdot \begin{bmatrix} 2 & 4 \\ 8 & 3 \\ 1 & 9 \\ 0 & 3 \end{bmatrix}
   = \begin{bmatrix} 14 & 28 \\ 56 & 21 \\ 7 & 63 \\ 0 & 21 \end{bmatrix}.
   \]

2. Compute \( 2B \):
   \[
   2B = 2 \cdot \begin{bmatrix} 8 & 4 \\ 1 & 2 \\ -1 & 3 \\ -4 & 1 \end{bmatrix}
   = \begin{bmatrix} 16 & 8 \\ 2 & 4 \\ -2 & 6 \\ -8 & 2 \end{bmatrix}.
   \]

3. Compute \( 7A - 2B \):
   \[
   7A - 2B = \begin{bmatrix} 14 & 28 \\ 56 & 21 \\ 7 & 63 \\ 0 & 21 \end{bmatrix}
   - \begin{bmatrix} 16 & 8 \\ 2 & 4 \\ -2 & 6 \\ -8 & 2 \end{bmatrix}
   = \begin{bmatrix} -2 & 20 \\ 54 & 17 \\ 9 & 57 \\ 8 & 19 \end{bmatrix}.
   \]

### Final Answer:
\[
7A - 2B = \begin{bmatrix} -2 & 20 \\ 54 & 17 \\ 9 & 57 \\ 8 & 19 \end{bmatrix}.
\]

Let me know if you would like a deeper delve into any step!

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**Relative Questions:**
1. How do scalar multiplications affect the dimensions of a matrix?
2. Can you apply matrix operations like this to non-square matrices?
3. What are the properties of matrix subtraction?
4. How would the calculation change if \( A \) and \( B \) had mismatched dimensions?
5. Why is order important in matrix operations?

**Tip:** Always verify matrix dimensions before performing operations to avoid errors!

Use a graphing utility to perform the indicated operations on the matrices given below. Find 7A - 2B, where A = [[2, 4], [8, 3], [1, 9], [0, 3]] and B = [[8, 4], [1, 2], [-1, 3], [-4, 1]].

Learn how to perform matrix operations to calculate 7A - 2B for given matrices A and B. This problem includes scalar multiplication and matrix subtraction, providing step-by-step calculations suitable for Grades 10-12.

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