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.