1. **Stating the problem:** We have a transformation matrix $$\begin{bmatrix} b & -2 \\ 16 & c \end{bmatrix}$$ that maps the point $$\begin{bmatrix} 5 \\ 1 \end{bmatrix}$$ onto the point $$\begin{bmatrix} -1 \\ 3 \end{bmatrix}$$. We need to find the values of $b$ and $c$. 2. **Formula and explanation:** The transformation is given by matrix multiplication: $$\begin{bmatrix} b & -2 \\ 16 & c \end{bmatrix} \begin{bmatrix} 5 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 3 \end{bmatrix}$$ This means: $$\begin{cases} 5b + (-2)(1) = -1 \\ 16(5) + c(1) = 3 \end{cases}$$ 3. **Solve the system:** - From the first equation: $$5b - 2 = -1$$ $$5b = 1$$ $$b = \frac{1}{5}$$ - From the second equation: $$80 + c = 3$$ $$c = 3 - 80$$ $$c = -77$$ 4. **Final answer:** $$b = \frac{1}{5}, \quad c = -77$$