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Without computing them, prove that the eigenvalues of the matrix $$ A=\left[\begin{array}{rrr} 6 & 2 & 1 \\ 1 & -5 & 0 \\ 2 & 1 & 4 \end{arr...

Question

Without computing them, prove that the eigenvalues of the matrix $$ A=\left[\begin{array}{rrr} 6 & 2 & 1 \\ 1 & -5 & 0 \\ 2 & 1 & 4 \end{array}\right] $$ satisfy the inequality $1 \leq|\lambda| \leq 9$.

Without computing them, prove that the eigenvalues of the matrix $$ A=\left[\begin{array}{rrr} 6 & 2 & 1 \\ 1 & -5 & 0 \\ 2 & 1 & 4 \end{array}\right] $$ satisfy the inequality $1 \leq|\lambda| \leq 9$.



Answers

Let $\lambda$ be any eigenvalue of a symmetric matrix $A .$ Justify the statement made in this section that $m \leq \lambda \leq M,$ where $m$ and $M$ are defined as in $(2) .[\text { Hint: Find an } \mathbf{x} \text { such that }$ $\lambda=\mathbf{x}^{T} A \mathbf{x} . ]$


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