https://file.aiquickdraw.com/mathbot/file/1730469383770skxhuvm4.jpg

Let's start by analyzing the problem and substituting the values given.

### Step 1: Substitute the values for \( a \), \( b \), and \( c \).
Given that:
- \( a = 5 \)
- \( b = 7 \)
- \( c = 8 \)

### Step 2: Write out the matrices with these values.

1. **Matrix A:**
   \[
   A = \begin{bmatrix} -1 & c+1 & 2 \\ 2 & 3 & -3 \\ 4 & 6 & 4 \end{bmatrix} = \begin{bmatrix} -1 & 8+1 & 2 \\ 2 & 3 & -3 \\ 4 & 6 & 4 \end{bmatrix} = \begin{bmatrix} -1 & 9 & 2 \\ 2 & 3 & -3 \\ 4 & 6 & 4 \end{bmatrix}
   \]

2. **Matrix B:**
   \[
   B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & a+b \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 5+7 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 12 \end{bmatrix}
   \]

3. **Matrix C:**
   \[
   C = \begin{bmatrix} 1 & -2 \\ -7 & 6 \\ 3 & -1 \end{bmatrix}
   \]

### Step 3: Calculate \( B - C \).

Subtracting matrix \( C \) from matrix \( B \):
\[
B - C = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 12 \end{bmatrix} - \begin{bmatrix} 1 & -2 \\ -7 & 6 \\ 3 & -1 \end{bmatrix} = \begin{bmatrix} 1 - 1 & 2 - (-2) \\ 3 - (-7) & 4 - 6 \\ 5 - 3 & 12 - (-1) \end{bmatrix}
\]

Simplifying the elements:
\[
B - C = \begin{bmatrix} 0 & 4 \\ 10 & -2 \\ 2 & 13 \end{bmatrix}
\]

### Step 4: Calculate \( A(B - C) \).

Now, we perform the matrix multiplication \( A \times (B - C) \), where \( A \) is a \( 3 \times 3 \) matrix and \( B - C \) is a \( 3 \times 2 \) matrix. The result will be a \( 3 \times 2 \) matrix.

\[
A(B - C) = \begin{bmatrix} -1 & 9 & 2 \\ 2 & 3 & -3 \\ 4 & 6 & 4 \end{bmatrix} \times \begin{bmatrix} 0 & 4 \\ 10 & -2 \\ 2 & 13 \end{bmatrix}
\]

Calculating each element:

1. **First row:**
   \[
   \begin{aligned}
   &(-1 \times 0) + (9 \times 10) + (2 \times 2) = 0 + 90 + 4 = 94, \\
   &(-1 \times 4) + (9 \times -2) + (2 \times 13) = -4 - 18 + 26 = 4.
   \end{aligned}
   \]

2. **Second row:**
   \[
   \begin{aligned}
   &(2 \times 0) + (3 \times 10) + (-3 \times 2) = 0 + 30 - 6 = 24, \\
   &(2 \times 4) + (3 \times -2) + (-3 \times 13) = 8 - 6 - 39 = -37.
   \end{aligned}
   \]

3. **Third row:**
   \[
   \begin{aligned}
   &(4 \times 0) + (6 \times 10) + (4 \times 2) = 0 + 60 + 8 = 68, \\
   &(4 \times 4) + (6 \times -2) + (4 \times 13) = 16 - 12 + 52 = 56.
   \end{aligned}
   \]

### Final Result:
\[
A(B - C) = \begin{bmatrix} 94 & 4 \\ 24 & -37 \\ 68 & 56 \end{bmatrix}
\]

---

Would you like further details on matrix multiplication or other concepts? 

Here are some related questions to deepen your understanding:

1. How do you determine the dimensions of the resulting matrix in matrix multiplication?
2. What are some common properties of matrix operations?
3. How can we apply matrix operations to solve systems of linear equations?
4. What are the conditions for two matrices to be added or multiplied?
5. How does changing the order of matrix multiplication affect the result?

**Tip:** In matrix multiplication, each element in the result matrix is derived from the sum of the products of elements across the row of the first matrix and down the column of the second matrix.

Given matrices A, B, and C with values depending on a, b, and c, determine the value of A(B - C) for a = 5, b = 7, and c = 8.

Learn how to determine the value of A(B - C) with matrices A, B, and C that depend on values a = 5, b = 7, and c = 8. This solution covers matrix subtraction and multiplication operations, step-by-step. Ideal for college-level linear algebra students.

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