https://file.aiquickdraw.com/mathbot/file/172606014875444m6zqjm.jpg

Let's analyze the two parts of the problem and solve them for \( k \).

### Part (a):
The equation in part (a) is given as:

\[
[k \ 1 \ 1] \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 2 & -2 \end{bmatrix} \begin{bmatrix} k \\ 1 \\ 1 \end{bmatrix} = 0
\]

To solve this, we follow these steps:

1. **Matrix multiplication:** First, compute the matrix product of the 3x3 matrix with the column vector \(\begin{bmatrix} k \\ 1 \\ 1 \end{bmatrix}\).

2. **Multiply by the row vector:** After getting the resulting vector from step 1, multiply it by \([k \ 1 \ 1]\).

3. **Set the result to 0:** Solve for \( k \) after simplifying the resulting equation.

### Part (b):
The equation in part (b) is:

\[
[3 \ 3 \ k] \begin{bmatrix} 1 & 2 & 0 \\ 1 & 1 & 2 \\ 4 & 0 & 1 \end{bmatrix} \begin{bmatrix} 3 \\ 3 \\ k \end{bmatrix} = 0
\]

Similar to part (a), the steps are:

1. **Matrix multiplication:** Multiply the given matrix by the column vector \(\begin{bmatrix} 3 \\ 3 \\ k \end{bmatrix}\).

2. **Multiply by the row vector:** Multiply the resulting vector by \([3 \ 3 \ k]\).

3. **Solve for \( k \):** Set the equation equal to 0 and solve for \( k \).

I will now solve both parts to find the values of \( k \).### Solutions:

- **Part (a):** The values of \( k \) are complex:
  \[
  k = -\frac{1}{2} - \frac{\sqrt{7}}{2}i \quad \text{or} \quad k = -\frac{1}{2} + \frac{\sqrt{7}}{2}i
  \]

- **Part (b):** The real solutions for \( k \) are:
  \[
  k = -15 \quad \text{or} \quad k = -3
  \]

Would you like further details on any of the steps or have additional questions?

Here are 5 related questions for deeper exploration:

1. How can you check if a system of linear equations has complex solutions?
2. What does it mean when matrix multiplication yields complex numbers?
3. What role does eigenvalues play in solving these types of matrix-vector problems?
4. How can we interpret the geometric meaning of solutions to matrix equations?
5. How does one handle complex solutions in physical or real-world contexts?

**Tip:** When dealing with matrices, always verify whether your matrix is symmetric, diagonalizable, or invertible, as this can simplify solving the system.

Find all values of k, if any, that satisfies the following equations (involving matrix operations).

Solve for k in two matrix equations using linear algebra concepts such as matrix multiplication and homogeneous systems. This problem is suitable for undergraduate students studying linear algebra.

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