1. **Problem statement:** Given matrix $A = \begin{pmatrix} 2 & 7 \\ 4 & k \end{pmatrix}$ and an eigenvector $V_1 = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$, find the value of $k$ and the corresponding eigenvalue. 2. **Recall the eigenvalue equation:** $$A V = \lambda V$$ where $\lambda$ is the eigenvalue and $V$ is the eigenvector. 3. **Apply the equation:** $$A V_1 = \lambda V_1$$ Substitute $A$ and $V_1$: $$\begin{pmatrix} 2 & 7 \\ 4 & k \end{pmatrix} \begin{pmatrix} 1 \\ 4 \end{pmatrix} = \lambda \begin{pmatrix} 1 \\ 4 \end{pmatrix}$$ 4. **Calculate the left side:** $$\begin{pmatrix} 2 \times 1 + 7 \times 4 \\ 4 \times 1 + k \times 4 \end{pmatrix} = \begin{pmatrix} 2 + 28 \\ 4 + 4k \end{pmatrix} = \begin{pmatrix} 30 \\ 4 + 4k \end{pmatrix}$$ 5. **Set equal to right side:** $$\begin{pmatrix} 30 \\ 4 + 4k \end{pmatrix} = \begin{pmatrix} \lambda \\ 4 \lambda \end{pmatrix}$$ 6. **From the first component:** $$30 = \lambda$$ 7. **From the second component:** $$4 + 4k = 4 \lambda = 4 \times 30 = 120$$ 8. **Solve for $k$:** $$4k = 120 - 4 = 116 \implies k = \frac{116}{4} = 29$$ 9. **Answer:** $$k = 29, \quad \lambda = 30$$