3. Social media analytics often involves understanding the interactions among different types of engagement metrics, such as likes, shares, and comments. In this model, the matrix \( A \) represents the equilibrium state of interactions across these metrics, where each entry reflects how one type of engagement influences another. For instance, an increase in likes might reduce shares or comments, simulating a balanced engagement system. The matrix \( A \) can be used to analyse long-term patterns and equilibrium states in engagement types, helping social media platforms understand stable configurations of user interactions. For this question, consider the matrix \( A \) given by:
\[
A=\left(\begin{array}{ccc}
-6 & 5 & -1 \\
-4 & 3 & -1 \\
0 & 6 & 4
\end{array}\right)
\]
(a) Find the characteristic polynomial of \( A \).
(b) The Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic polynomial. That is, for a matrix \( A \) with characteristic polynomial \( \chi_{A}(\lambda)= \) \( a_{3} \lambda^{3}+a_{2} \lambda^{2}+a_{1} \lambda+a_{0} \), the theorem implies that:
\[
a_{3} A^{3}+a_{2} A^{2}+a_{1} A+a_{0} I=0 .
\]
In the context of social media analytics, this means that \( A \), which represents interactions among engagement metrics, behaves in a predictable way governed by its characteristic polynomial. By applying the Cayley-Hamilton theorem, we can understand how repeated interactions among engagement types stabilize over time.
For the matrix in part a), compute \( A^{2} \) and \( A^{3} \). Hence validate that the matrix \( A \) satisfies its own characteristic polynomial
(c) Using the Cayley-Hamilton Theorem on the given matrix \( A \), write \( A^{3} \) as a linear combination of \( I, A, A^{2} \). Hence use this result to express \( A^{4}, A^{5} \), and \( A^{6} \) in terms of \( I, A \), and \( A^{2} \).
(d) Using the Cayley-Hamilton Theorem on the given matrix \( A \), isolate the identity matrix from the equation and hence find the inverse of \( A \) algebraically as a linear combination of \( I, A, A^{2} \) and numerically as a matrix.
