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.