https://file.aiquickdraw.com/mathbot/file/1726419716200686czb8v.jpg

Given matrices:
\[
A = \begin{bmatrix} 3 & 6 \\ 4 & x \end{bmatrix}, \quad B = \begin{bmatrix} 5 & -2 \\ 3 & 4 \end{bmatrix}
\]
and the problem requires finding \( x \) such that:
\[
\det(5A^{-1}B^T) = 5
\]

### Step-by-Step Solution

1. **Transpose of Matrix \( B \):**
   \[
   B^T = \begin{bmatrix} 5 & 3 \\ -2 & 4 \end{bmatrix}
   \]

2. **Determinant of Matrix \( A \):**
   The determinant of \( A \) is:
   \[
   \det(A) = (3 \times x) - (4 \times 6) = 3x - 24
   \]
   For \( A \) to be invertible, \(\det(A) \neq 0\), so \(3x - 24 \neq 0\). Thus, \( x \neq 8 \).

3. **Inverse of Matrix \( A \):**
   The inverse of a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by:
   \[
   A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
   \]
   Applying this formula:
   \[
   A^{-1} = \frac{1}{3x - 24} \begin{bmatrix} x & -6 \\ -4 & 3 \end{bmatrix}
   \]

4. **Determinant of \( 5A^{-1}B^T \):**
   Using the property of determinants, \(\det(kM) = k^n \det(M)\) for an \( n \times n \) matrix \( M \):
   \[
   \det(5A^{-1}B^T) = 5^2 \det(A^{-1}) \det(B^T)
   \]
   Since \(\det(A^{-1}) = \frac{1}{\det(A)}\) and \(\det(B^T) = \det(B)\) (the determinant of a matrix is the same as its transpose):
   \[
   \det(5A^{-1}B^T) = 25 \times \frac{1}{3x - 24} \times \det(B)
   \]

5. **Determinant of Matrix \( B \):**
   \[
   \det(B) = (5 \times 4) - (3 \times -2) = 20 + 6 = 26
   \]

6. **Setting up the equation:**
   \[
   25 \times \frac{1}{3x - 24} \times 26 = 5
   \]

7. **Solving for \( x \):**
   \[
   \frac{650}{3x - 24} = 5
   \]
   Multiply both sides by \( 3x - 24 \):
   \[
   650 = 5(3x - 24)
   \]
   Divide by 5:
   \[
   130 = 3x - 24
   \]
   Add 24 to both sides:
   \[
   154 = 3x
   \]
   Divide by 3:
   \[
   x = \frac{154}{3} \approx 51.\overline{3}
   \]

Thus, the value of \( x \) that satisfies the equation is \( x = \frac{154}{3} \).

Would you like further details or have any questions about this problem?

### Follow-Up Questions

1. What are the properties of determinants used in this solution?
2. How do you find the inverse of a general 2x2 matrix?
3. Why is the determinant of a matrix important in finding its invertibility?
4. How does the property \(\det(kA) = k^n \det(A)\) work for matrices of different dimensions?
5. What happens if the determinant of matrix \( A \) is zero?

### Tip

When dealing with matrix inverses, always ensure the determinant is non-zero, as a zero determinant indicates the matrix is singular (non-invertible).

Given matrices A = [ [3, 6], [4, x] ] and B = [ [5, -2], [3, 4] ], find the value of x such that det(5A^(-1) * B^T) = 5.

Learn how to solve a matrix algebra problem involving inverses, transposes, and determinants. Given matrices A and B, find the value of x that satisfies the equation det(5A^(-1) * B^T) = 5. This problem covers matrix determinants, matrix inverses, and key matrix properties, suitable for students in grades 11-12 and college-level mathematics.

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