1. **State the problem:** Find all values of $k$ such that the matrix-vector products equal the zero vector in the given equations. 2. **Exercise 15:** Given matrix $A = \begin{bmatrix} k & 1 & 1 \\ 1 & 0 & 2 \\ 0 & 2 & -3 \end{bmatrix}$ and vector $\mathbf{v} = \begin{bmatrix} k \\ 1 \\ 1 \end{bmatrix}$, solve $A\mathbf{v} = \mathbf{0}$. 3. **Write the matrix multiplication:** $$ \begin{bmatrix} k & 1 & 1 \\ 1 & 0 & 2 \\ 0 & 2 & -3 \end{bmatrix} \begin{bmatrix} k \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} k \cdot k + 1 \cdot 1 + 1 \cdot 1 \\ 1 \cdot k + 0 \cdot 1 + 2 \cdot 1 \\ 0 \cdot k + 2 \cdot 1 + (-3) \cdot 1 \end{bmatrix} = \begin{bmatrix} k^2 + 2 \\ k + 2 \\ 2 - 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} $$ 4. **Set up equations:** $$ \begin{cases} k^2 + 2 = 0 \\ k + 2 = 0 \\ 2 - 3 = 0 \end{cases} $$ 5. **Analyze the third equation:** $2 - 3 = -1 \neq 0$, so the system has no solution for any $k$. 6. **Conclusion for Exercise 15:** No value of $k$ satisfies the equation because the last component is always $-1$. --- 7. **Exercise 16:** Given matrix $B = \begin{bmatrix} 2 & 2 & k \\ 2 & 0 & 3 \\ 0 & 3 & 1 \end{bmatrix}$ and vector $\mathbf{w} = \begin{bmatrix} 2 \\ 2 \\ k \end{bmatrix}$, solve $B\mathbf{w} = \mathbf{0}$. 8. **Write the matrix multiplication:** $$ \begin{bmatrix} 2 & 2 & k \\ 2 & 0 & 3 \\ 0 & 3 & 1 \end{bmatrix} \begin{bmatrix} 2 \\ 2 \\ k \end{bmatrix} = \begin{bmatrix} 2 \cdot 2 + 2 \cdot 2 + k \cdot k \\ 2 \cdot 2 + 0 \cdot 2 + 3 \cdot k \\ 0 \cdot 2 + 3 \cdot 2 + 1 \cdot k \end{bmatrix} = \begin{bmatrix} 4 + 4 + k^2 \\ 4 + 3k \\ 6 + k \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} $$ 9. **Set up equations:** $$ \begin{cases} 8 + k^2 = 0 \\ 4 + 3k = 0 \\ 6 + k = 0 \end{cases} $$ 10. **Solve the third equation:** $$k = -6$$ 11. **Check the second equation with $k = -6$:** $$4 + 3(-6) = 4 - 18 = -14 \neq 0$$ 12. **Check the first equation with $k = -6$:** $$8 + (-6)^2 = 8 + 36 = 44 \neq 0$$ 13. **Conclusion for Exercise 16:** No value of $k$ satisfies all three equations simultaneously. **Final answer:** No values of $k$ satisfy the given matrix equations in Exercises 15 and 16.