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5. (10 Points) Suppose we have the following coefficient matrix: \( A=\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 4 & 5 \\ 0 & 4 & 3\end{array}\right) \). We will answer some questions related to its eigenvalues and eigenvectors. NOTE: You are not calculating eigenvalues and eigenvectors here. Instead, we will demonstrate how we can assess whether or not a value or vector is an eigenvalue or eigenvector. a.) Suppose I have found an eigenvector. Geometrically what is so special about this vector in comparison to some other arbitrary vector? b.) How do you know that \( \left(\begin{array}{c}2 \\ -2 \\ 6\end{array}\right) \) is not an eigenvector of this matrix? c) Show that one of the eigenvalues is - 1 . That is, demonstrate that you can find a general eigenvector. Give 3 possible eigenvectors. d) Show that 4 is not an eigenvalue of this matrix.

Geometrically an Eigen vectors c

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