Let $A$ be a $2\times2$ matrix with integer entries satisfying $A^{2}=3A-2I$, where $I$ is the identity matrix. (i) Determine $\operatorname{tr}(A)$ and $\det(A)$. (ii) Express $A^{5}$ in the form $pA+qI$ and give the integers $p$ and $q$.

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tr(A)=3, det(A)=2; A^5 = 31A - 30I

Gce Model Questions Vol1 2022 Further Mathematics Question 3

Let \(A\) be a \(2\times2\) matrix with integer entries satisfying \(A^{2}=3A-2I\), where \(I\) is the identity matrix. (i) Determine \(\operatorname{tr}(A)\) and \(\det(A)\). (ii) Express \(A^{5}\) in the form \(pA+qI\) and give the integers \(p\) and \(q\).

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