1. **Problem Statement:** Find the limit as $x \to \infty$ of the product of nine identical $2 \times 2$ matrices: $$\lim_{x \to \infty} \left(\begin{bmatrix}a & b \\ c & d\end{bmatrix}^9\right)$$ 2. **Understanding the Problem:** We are asked to find the limit of the matrix raised to the 9th power as $x$ approaches infinity. However, the matrix does not depend on $x$, so the limit is simply the matrix to the 9th power. 3. **Key Formula:** Matrix exponentiation for integer powers is defined as repeated multiplication: $$A^n = \underbrace{A \times A \times \cdots \times A}_{n \text{ times}}$$ where $A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$. 4. **Important Notes:** - Since the matrix is constant (no $x$ dependence), the limit as $x \to \infty$ is just $A^9$. - To compute $A^9$, one can use diagonalization if $A$ is diagonalizable, or Jordan normal form otherwise. 5. **Intermediate Work:** - Find eigenvalues $\lambda$ by solving: $$\det(A - \lambda I) = 0 \Rightarrow (a - \lambda)(d - \lambda) - bc = 0$$ - If $A$ is diagonalizable, write $A = PDP^{-1}$ where $D$ is diagonal with eigenvalues. - Then: $$A^9 = PD^9P^{-1}$$ where $$D^9 = \begin{bmatrix}\lambda_1^9 & 0 \\ 0 & \lambda_2^9\end{bmatrix}$$ 6. **Conclusion:** The limit is the matrix $A$ raised to the 9th power, which can be computed via diagonalization or repeated multiplication. **Final answer:** $$\lim_{x \to \infty} A^9 = A^9$$ This is a constant matrix independent of $x$.