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).