https://file.aiquickdraw.com/mathbot/file/1726483691213vyh8v3kw.jpg

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.

Let matrix A = [[-2, 0], [3, 1]] and matrix B = [[2k - 1, 1], [2, k + 1]], where k is a constant. Determine the value(s) of k if matrix B is singular.

Learn how to determine the value(s) of k that make the 2x2 matrix B singular, using the determinant and quadratic equation formula. Suitable for advanced high school algebra or pre-calculus students.

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