1. The problem is to find the product of three identical 2x2 matrices: $$\begin{bmatrix}a & b \\ c & d\end{bmatrix} \times \begin{bmatrix}a & b \\ c & d\end{bmatrix} \times \begin{bmatrix}a & b \\ c & d\end{bmatrix}$$. 2. The formula for matrix multiplication of two 2x2 matrices $$A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$$ and $$B = \begin{bmatrix}e & f \\ g & h\end{bmatrix}$$ is: $$AB = \begin{bmatrix}ae + bg & af + bh \\ ce + dg & cf + dh\end{bmatrix}$$. 3. First, compute $$M^2 = M \times M$$ where $$M = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$$: $$M^2 = \begin{bmatrix}a^2 + bc & ab + bd \\ ac + dc & bc + d^2\end{bmatrix}$$. 4. Next, compute $$M^3 = M^2 \times M$$: $$M^3 = \begin{bmatrix}(a^2 + bc)a + (ab + bd)c & (a^2 + bc)b + (ab + bd)d \\ (ac + dc)a + (bc + d^2)c & (ac + dc)b + (bc + d^2)d\end{bmatrix}$$. 5. Simplify each element: - Top-left: $$a^3 + abc + abc + bdc = a^3 + 2abc + bdc$$ - Top-right: $$a^2b + bcb + abd + b^2d = a^2b + b^2c + abd + b^2d$$ - Bottom-left: $$aca + dca + bcc + d^2c = a^2c + dca + bc^2 + d^2c$$ - Bottom-right: $$acb + dcb + bcd + d^3 = abc + dcb + bcd + d^3$$ 6. Final matrix: $$\begin{bmatrix}a^3 + 2abc + bdc & a^2b + b^2c + abd + b^2d \\ a^2c + dca + bc^2 + d^2c & abc + dcb + bcd + d^3\end{bmatrix}$$. This is the product of the three matrices.