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{ and } P_{\mathcal{A}}^{\mathcal{C}} \text { are the coor- }\\ &\text { dinate transition matrices from } \mathcal{A} \text { to } \mathcal{B}, \mathcal{C} \text { to } \mathcal{B}, \text { and } \mathcal{A} \text { to } \mathcal{C}, \text { respectively; i.e. } \forall x \in \mathbf{R}^{2} \text { , } \end{aligned} \begin{array}{l} {[x]_{B}=P_{A}^{B}[x]_{\mathcal{A}}} \\ {[x]_{B}=P_{C}^{B}[x]_{C}} \\ {[x]_{C}=P_{\mathcal{A}}^{\mathcal{C}}[x]_{\mathcal{A}}} \end{array} \text { Further suppose } P_{A}^{B}=\left[\begin{array}{ll} 4 & 3 \\ 3 & 2 \end{array}\right] \text { and } P_{C}^{B}=\left[\begin{array}{cc} 11 & 2 \\ 5 & 1 \end{array}\right] . \text { Find } P_{A}^{\mathcal{C}}, \text { and justify your answer. }

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