To solve this problem, we need to address two main points:

1. The matrix $B$ is given as: $B = \begin{bmatrix} 3x & x \\ 2x & \frac{x}{2} \end{bmatrix}$

2. We need to determine the possible values of $x$ such that the determinant $|B| = -8$ and $B$ is singular.

Steps to Solution

Step 1: Calculate the Determinant of $B$

For a $2 \times 2$ matrix of the form: $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ the determinant is calculated as: $|B| = ad - bc$

In this case:

• $a = 3x$
• $b = x$
• $c = 2x$
• $d = \frac{x}{2}$

So the determinant of $B$ is: $|B| = (3x) \left(\frac{x}{2}\right) - (x)(2x)$ Expanding this: $|B| = \frac{3x^2}{2} - 2x^2$

To simplify further, convert $2x^2$ to have a common denominator: $|B| = \frac{3x^2}{2} - \frac{4x^2}{2} = \frac{-x^2}{2}$ Thus: $|B| = -\frac{x^2}{2}$

Step 2: Set the Determinant to $-8$

Since we are given $|B| = -8$, substitute this into our equation: $-\frac{x^2}{2} = -8$

Multiply both sides by $-2$: $x^2 = 16$

Taking the square root of both sides, we get: $x = \pm 4$

Step 3: Check the Condition for $B$ to be Singular

A matrix is singular if its determinant is zero. However, since we are given a specific non-zero value for the determinant ($-8$), this means the condition of singularity here might be a misunderstanding, or they intended to highlight the calculation with $|B| = -8$ explicitly rather than enforcing singularity directly.

Final Answer

The possible values of $x$ are: $x = \pm 4$

Would you like a deeper explanation on any specific part of this solution?

Here are some additional questions to consider:

1. How would the solution change if the matrix $B$ were required to be singular?
2. What is the determinant when $x = 0$, and does it satisfy any conditions?
3. What is the significance of the matrix being singular in this problem?
4. How do we interpret complex solutions if any arise in a similar problem?
5. What other methods can be used to verify the determinant calculations?

Tip: When working with determinants, always check if a matrix can be simplified before computation for faster results.