Math Snap

STEP 1

1. The problem involves matrix operations including finding the inverse, transpose, and determinant.
2. The determinant of a matrix product can be expressed as the product of the determinants.
3. The inverse of a matrix $A$ exists if and only if $\det(A) \neq 0$.

STEP 2

1. Find the inverse of matrix $A$.
2. Find the transpose of matrix $B$.
3. Calculate the determinant of the product $5A^{-1}B^t$.
4. Solve for $x$.

STEP 3

Calculate the determinant of matrix $A$ to ensure it is invertible:
$$A = \begin{bmatrix} 3 & 6 \\ 4 & x \end{bmatrix}$$ The determinant of $A$ is given by:
$$\det(A) = 3x - 6 \times 4 = 3x - 24$$ For $A$ to be invertible, $\det(A) \neq 0$.

STEP 4

Assuming $\det(A) \neq 0$, find the inverse of $A$:
The inverse of a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is:
$$A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$ Thus, for matrix $A$:
$$A^{-1} = \frac{1}{3x - 24} \begin{bmatrix} x & -6 \\ -4 & 3 \end{bmatrix}$$

STEP 5

Find the transpose of matrix $B$:
$$B = \begin{bmatrix} 5 & -2 \\ 3 & 4 \end{bmatrix}$$ The transpose of $B$, denoted $B^t$, is:
$$B^t = \begin{bmatrix} 5 & 3 \\ -2 & 4 \end{bmatrix}$$

STEP 6

Calculate the determinant of the product $5A^{-1}B^t$.
Using the property of determinants, $\det(kA) = k^n \det(A)$ for a scalar $k$ and an $n \times n$ matrix $A$, and $\det(AB) = \det(A)\det(B)$:
$$\det(5A^{-1}B^t) = 5^2 \cdot \det(A^{-1}) \cdot \det(B^t)$$ Since $\det(A^{-1}) = \frac{1}{\det(A)}$ and $\det(B^t) = \det(B)$:
$$\det(5A^{-1}B^t) = 25 \cdot \frac{1}{3x - 24} \cdot \det(B)$$ Calculate $\det(B)$:
$$\det(B) = 5 \times 4 - (-2) \times 3 = 20 + 6 = 26$$ Thus:
$$\det(5A^{-1}B^t) = 25 \cdot \frac{26}{3x - 24}$$ Set the determinant equal to 5:
$$25 \cdot \frac{26}{3x - 24} = 5$$

Solve for $x$:
Simplify the equation:
$$\frac{650}{3x - 24} = 5$$ Multiply both sides by $3x - 24$:
$$650 = 5(3x - 24)$$ Expand and solve for $x$:
$$650 = 15x - 120$$ Add 120 to both sides:
$$770 = 15x$$ Divide by 15:
$$x = \frac{770}{15} = \frac{154}{3}$$ The value of $x$ is:
$$\boxed{\frac{154}{3}}$$