Let's first address the second part of the question:

Matrix $B$: $B = \begin{bmatrix} 2k - 1 & 1 \\ 2 & k + 1 \end{bmatrix}$ We are asked to find the value(s) of $k$ that make matrix $B$ singular.

Step 1: Definition of Singular Matrix

A matrix is singular if and only if its determinant is zero.

Step 2: Determinant of Matrix $B$

The determinant of a 2x2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is calculated as: $\text{det}(B) = (a \cdot d) - (b \cdot c)$ For matrix $B$: $\text{det}(B) = (2k - 1)(k + 1) - (1 \cdot 2)$

Step 3: Solve for the Determinant

Let's expand the determinant: $\text{det}(B) = (2k - 1)(k + 1) - 2$ Expanding the product: $\text{det}(B) = (2k^2 + 2k - k - 1) - 2$ Simplifying: $\text{det}(B) = 2k^2 + k - 1 - 2 = 2k^2 + k - 3$

Step 4: Set the Determinant to Zero

To make $B$ singular, set the determinant equal to zero: $2k^2 + k - 3 = 0$

Step 5: Solve the Quadratic Equation

We can solve this quadratic equation using the quadratic formula: $k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Where $a = 2$, $b = 1$, and $c = -3$.

Let's solve for $k$.The values of $k$ that make matrix $B$ singular are: $k = 1 \quad \text{or} \quad k = -1.5$

Would you like any further details or clarifications?

Here are 5 related questions:

1. How do you determine if a matrix is invertible?
2. Can you explain how the determinant of a matrix affects its singularity?
3. What is the geometric interpretation of a singular matrix?
4. How does changing $k$ affect the determinant of matrix $B$?
5. How can matrix $A$ in the problem be used in further calculations with matrix $B$?

Tip: For quadratic equations, always check the discriminant $b^2 - 4ac$ first. If it’s negative, the equation has no real solutions.