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The image you provided contains a matrix and a function-related problem. Here's the transcription of the content:

Matrix:
\[
\begin{pmatrix}
-7 & 3 & 6 \\
(8a - 23) & 7 & 1
\end{pmatrix}
\]

Problem Statement:
"Find the value of \(a\) so that the function \( f(a) = \text{det}(A) = 8a + b \) has a value of 4. (The result of the determinant is 7.)"

### Solution:

1. **Determinant of the Matrix**:
   The matrix \( A \) is:
   \[
   A = \begin{pmatrix}
   -7 & 3 & 6 \\
   (8a - 23) & 7 & 1
   \end{pmatrix}
   \]
   The determinant of a 2x3 matrix is calculated as:
   \[
   \text{det}(A) = \text{ad} - \text{bc}
   \]
   However, the provided matrix isn't square (2x3), so the direct determinant formula for 2x2 or 3x3 doesn't apply here. But the problem suggests a function relation where the determinant result should be 7.

2. **Function \( f(a) = \text{det}(A) = 8a + b \)**:
   According to the problem, \( f(a) = 8a + b \) equals 7 when \( \text{det}(A) = 4 \).

   To find \( a \), we set the function equal to 7:
   \[
   8a + b = 7
   \]

3. **Given Condition**:
   Since the problem states that \( \text{det}(A) = 4 \), we substitute into the function:
   \[
   4 = 8a + b
   \]

   Now we have two equations:
   1. \( 8a + b = 7 \)
   2. \( 8a + b = 4 \)

   These equations are contradictory unless \( 7 = 4 \), which isn't possible.

**Possible Clarification Needed:**
- There might be a misinterpretation in the problem statement or missing information since the determinant equation provided doesn't yield a solvable equation for \( a \). It might require a closer examination or correction of the problem setup.

Would you like more detailed steps on this or have any other questions? Here are some follow-up questions that might interest you:

1. How is the determinant calculated for non-square matrices?
2. What are the properties of the determinant that could help solve this problem?
3. How can we interpret the given matrix for a different context?
4. What does the function \( f(a) = 8a + b \) represent in terms of linear algebra?
5. How can we use systems of equations to solve for unknown variables like \( a \)?

**Tip:** Always ensure the matrix provided is square (same number of rows and columns) when asked to find a determinant, as the determinant is only defined for square matrices.

Find the value of a so that the function f(a) = det(A) = 8a + b has a value of 4. (The result of the determinant is 7.)

Explore the problem of finding the value of 'a' such that the function f(a) = det(A) = 8a + b equals 4. This problem involves matrix determinants, algebraic functions, and system of equations, suitable for students in Grades 10-12.

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