The equation given is:

$2 \begin{bmatrix} -1 \\ \frac{1}{2} \\ \frac{1}{2} \end{bmatrix} + 3 \begin{bmatrix} 4 \\ 0 \\ 3 \end{bmatrix} + k \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 2 \\ -3 \\ -4 \end{bmatrix}$

We need to solve for $k$. Let's first break this into manageable steps.

Step 1: Perform scalar multiplication for each matrix.

1. Multiply the first matrix by 2: $2 \begin{bmatrix} -1 \\ \frac{1}{2} \\ \frac{1}{2} \end{bmatrix} = \begin{bmatrix} -2 \\ 1 \\ 1 \end{bmatrix}$

2. Multiply the second matrix by 3: $3 \begin{bmatrix} 4 \\ 0 \\ 3 \end{bmatrix} = \begin{bmatrix} 12 \\ 0 \\ 9 \end{bmatrix}$

3. Multiply the third matrix by $k$: $k \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 2k \\ k \\ 3k \end{bmatrix}$

Step 2: Add the results from Step 1.

We now combine all the terms:

$\begin{bmatrix} -2 \\ 1 \\ 1 \end{bmatrix} + \begin{bmatrix} 12 \\ 0 \\ 9 \end{bmatrix} + \begin{bmatrix} 2k \\ k \\ 3k \end{bmatrix} = \begin{bmatrix} 2 \\ -3 \\ -4 \end{bmatrix}$

Step 3: Solve for $k$ by comparing the elements.

We can now form equations for each element:

1. First row: $-2 + 12 + 2k = 2$ $10 + 2k = 2 \implies 2k = -8 \implies k = -4$

2. Second row: $1 + 0 + k = -3$ $1 + k = -3 \implies k = -4$

3. Third row: $1 + 9 + 3k = -4$ $10 + 3k = -4 \implies 3k = -14 \implies k = -\frac{14}{3}$

Conclusion:

There's an inconsistency between the second and third row equations. Would you like me to check for any possible issues in the problem setup or explore alternate interpretations? Feel free to clarify if there's anything more specific you'd like to check!

Questions for expansion:

1. How is matrix addition and scalar multiplication applied to larger matrices?
2. Could we resolve the inconsistency by rechecking the given values?
3. How are simultaneous equations handled when dealing with matrices?
4. How would the solution change if the final matrix had different values?
5. What are the typical applications of solving matrix equations like these?

Tip: When solving matrix equations, always verify that each row aligns with the respective components to avoid inconsistencies.