To determine the value of $x$ when matrix $B$ is singular, we use the property that a matrix is singular if and only if its determinant is zero. The matrix $B$ is given as:

$B = \begin{bmatrix} -6 & 3 \\ 2x & 8 \end{bmatrix}$

Step 1: Calculate the Determinant

The determinant of a 2x2 matrix $B = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is:

$\text{det}(B) = ad - bc$

Applying this formula to the matrix $B$:

$\text{det}(B) = (-6)(8) - (3)(2x)$

$\text{det}(B) = -48 - 6x$

Step 2: Set the Determinant to Zero

Since the matrix is singular:

$-48 - 6x = 0$

Step 3: Solve for $x$

$-6x = 48$

$x = -8$

Conclusion

The value of $x$ that makes the matrix $B$ singular is $x = -8$.

Would you like more details or have any further questions?

Related Questions:

1. What does it mean for a matrix to be singular in terms of its rank?
2. How do you calculate the determinant of a 3x3 matrix?
3. What is the significance of the determinant in solving systems of linear equations?
4. How can you determine if a matrix is invertible using its determinant?
5. What is the geometric interpretation of a matrix having a determinant of zero?

Tip: Always check the determinant of a matrix when analyzing its properties, such as singularity or invertibility.