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We are given two matrices \( A \) and \( B \) as follows:

\[
A = \begin{bmatrix} 3 & 6 \\ 4 & x \end{bmatrix}, \quad B = \begin{bmatrix} 5 & -2 \\ 3 & 4 \end{bmatrix}
\]

The task is to find the determinant of \( 5A^{-1} B^t \), where the determinant is equal to 5.

### Step 1: Using properties of determinants

We know the following property of determinants:
\[
\text{det}(k A) = k^n \cdot \text{det}(A)
\]
where \( k \) is a scalar and \( A \) is an \( n \times n \) matrix. Here, the matrix \( A \) is \( 2 \times 2 \), so:
\[
\text{det}(5A^{-1} B^t) = 5^2 \cdot \text{det}(A^{-1}) \cdot \text{det}(B^t)
\]
Since the determinant of the transpose of a matrix is the same as the determinant of the matrix itself:
\[
\text{det}(5A^{-1} B^t) = 25 \cdot \text{det}(A^{-1}) \cdot \text{det}(B)
\]

Also, we know that:
\[
\text{det}(A^{-1}) = \frac{1}{\text{det}(A)}
\]
Thus, the equation becomes:
\[
\text{det}(5A^{-1} B^t) = 25 \cdot \frac{1}{\text{det}(A)} \cdot \text{det}(B)
\]

### Step 2: Determinant of \( B \)

First, we calculate the determinant of matrix \( B \):
\[
\text{det}(B) = (5)(4) - (-2)(3) = 20 + 6 = 26
\]

### Step 3: Determinant of \( A \)

We don't yet know \( x \), so we calculate the determinant of matrix \( A \) symbolically:
\[
\text{det}(A) = (3)(x) - (6)(4) = 3x - 24
\]

### Step 4: Substituting into the determinant equation

Substitute the determinant values into the equation:
\[
5 = 25 \cdot \frac{26}{3x - 24}
\]
Solve for \( x \):
\[
5 = \frac{650}{3x - 24}
\]
Multiplying both sides by \( 3x - 24 \) to eliminate the fraction:
\[
5(3x - 24) = 650
\]
Expanding the left-hand side:
\[
15x - 120 = 650
\]
Add 120 to both sides:
\[
15x = 770
\]
Finally, divide by 15:
\[
x = \frac{770}{15} = 51.33
\]

Thus, \( x = \frac{770}{15} \) or approximately \( x \approx 51.33 \).

Would you like more details on any step, or do you have any further questions?

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**5 Related Questions:**
1. How do you calculate the inverse of a 2x2 matrix?
2. What are the properties of matrix transposition in determinants?
3. Can the determinant of a matrix ever be negative, and what does it imply?
4. How can we find the eigenvalues of matrix \( A \) or \( B \)?
5. Why is the determinant of \( A^{-1} \) the reciprocal of the determinant of \( A \)?

**Tip:** When solving matrix equations involving determinants, using properties like \(\det(A^{-1}) = \frac{1}{\det(A)}\) can simplify calculations significantly.

Find the determinant of 5A^(-1) B^t, where A = [[3, 6], [4, x]] and B = [[5, -2], [3, 4]] such that the determinant equals 5.

Explore how to calculate the determinant of 5A^(-1) B^t given matrices A and B, where A = [[3, 6], [4, x]] and B = [[5, -2], [3, 4]]. This problem involves matrix algebra, including the use of determinant properties, inverse matrices, and matrix transposition. The solution includes a step-by-step guide to solve for x where the determinant equals 5.

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