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Scaling matrix powers to converge

University · ●●●○○ 50 pts

Let A=(1−10−12−10−11)A=\begin{pmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{pmatrix}. Find the real number aa for which the limit lim⁡n→∞anAn\lim_{n\to\infty} a^nA^n exists and is non-zero, where a sequence of matrices BnB_n converges to BB when Bnx→BxB_nx\to Bx for every x∈R3x\in\mathbb{R}^3. Give aa rounded to two decimal places.

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matrix powers