https://file.aiquickdraw.com/mathbot/file/1727432656196ov2k8wdm.jpg

Let's analyze the problem presented in the image step by step.

You are asked to find the value of \( x \) given that:
\[
A = \begin{bmatrix} 3 & 6 \\ 4 & x \end{bmatrix}, \quad B = \begin{bmatrix} 5 & -2 \\ 3 & 4 \end{bmatrix}
\]
and
\[
\text{det}\left(5A^{-1}B^t\right) = 5
\]

### Step-by-Step Solution

1. **Transpose of Matrix \( B \)**: 
   The transpose of matrix \( B \), denoted \( B^t \), is obtained by switching the rows and columns of \( B \):
   \[
   B^t = \begin{bmatrix} 5 & 3 \\ -2 & 4 \end{bmatrix}
   \]

2. **Determinant of a Product of Matrices**:
   The determinant of a product of matrices follows the rule:
   \[
   \text{det}(5A^{-1}B^t) = 5^2 \cdot \text{det}(A^{-1}) \cdot \text{det}(B^t)
   \]
   Since the determinant of a matrix inverse is the reciprocal of the determinant of the matrix, we can rewrite this as:
   \[
   5^2 \cdot \frac{1}{\text{det}(A)} \cdot \text{det}(B^t)
   \]
   We know this equals 5, so:
   \[
   25 \cdot \frac{1}{\text{det}(A)} \cdot \text{det}(B^t) = 5
   \]
   Simplifying this equation:
   \[
   \frac{\text{det}(B^t)}{\text{det}(A)} = \frac{1}{5}
   \]

3. **Determinant of \( A \)**:
   The determinant of matrix \( A \) is given by:
   \[
   \text{det}(A) = (3)(x) - (6)(4) = 3x - 24
   \]

4. **Determinant of \( B^t \)**:
   Since the determinant of the transpose of a matrix is the same as the determinant of the original matrix, we can directly find:
   \[
   \text{det}(B^t) = \text{det}(B) = (5)(4) - (-2)(3) = 20 + 6 = 26
   \]

5. **Final Equation**:
   Substitute the determinants into the equation:
   \[
   \frac{26}{3x - 24} = \frac{1}{5}
   \]
   Cross-multiply:
   \[
   26 \cdot 5 = 3x - 24
   \]
   Simplifying:
   \[
   130 = 3x - 24
   \]
   Add 24 to both sides:
   \[
   154 = 3x
   \]
   Finally, solve for \( x \):
   \[
   x = \frac{154}{3} \approx 51.33
   \]

Thus, the value of \( x \) is approximately \( 51.33 \).

Let me know if you'd like more details on any step or if you have any questions.

---

### Related Questions:
1. How is the determinant of a 2x2 matrix calculated?
2. What properties of determinants are used when solving matrix equations?
3. How do you calculate the inverse of a matrix?
4. Why is the transpose of a matrix relevant in this problem?
5. What is the significance of the determinant being non-zero?

### Tip:
When working with matrices, it's crucial to remember that the determinant helps determine if a matrix is invertible. A matrix with a zero determinant is singular and doesn't have an inverse.

Find det(5A^{-1}B^t) = 5, where A = [[3, 6], [4, x]] and B = [[5, -2], [3, 4]]

Step-by-step solution for finding x in the matrix equation det(5A^{-1}B^t) = 5, where A and B are 2x2 matrices. Learn how to compute determinants, matrix transposition, and inverse matrices. Ideal for undergraduate students.

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