1. The problem asks to determine values for $A$ and $B$ such that the system of equations $$\begin{cases} 12x + Ay = 8 \\ Ax + By = 4 \end{cases}$$ has infinitely many solutions. 2. For a system of two linear equations to have infinitely many solutions, the two equations must be dependent, meaning one is a scalar multiple of the other. 3. This means the ratios of the coefficients of $x$, $y$, and the constants must be equal: $$\frac{12}{A} = \frac{A}{B} = \frac{8}{4}$$ 4. Simplify the constant ratio: $$\frac{8}{4} = 2$$ 5. Set the first ratio equal to 2: $$\frac{12}{A} = 2 \implies 12 = 2A \implies A = 6$$ 6. Set the second ratio equal to 2: $$\frac{A}{B} = 2 \implies \frac{6}{B} = 2 \implies 6 = 2B \implies B = 3$$ 7. Therefore, the values are: $$A = 6, \quad B = 3$$ These values make the second equation a multiple of the first, so the system has infinitely many solutions.