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$B=A^{-1}=\begin{pmatrix}2&-1&0\\ -1&2&-1\\ 0&-1&1\end{pmatrix}$

What are the bound for $A$? You are not allowed to calculate the eigenvalues of $A$.

So for the matrix $B$, we have that it's trace is 5 and it's determinante is 1.

So $\lambda _1+\lambda _2\:+\lambda _3\:=5$ and $\lambda _1\lambda _2\:\lambda _3\:=1$

So first of is there a relation between the trace of a matrix and it's inverse, same question for the determinante? Also I do know that I can calculate the eigenvalues of the inverse and find the eigenvalues of the original matrix with that, but it seems to me that there is another way to find the bound without direct calculation.

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    Maybe https://en.wikipedia.org/wiki/Gershgorin_circle_theorem? – 2017-02-06
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    I think the question is trying to get you to apply a certain kind of bound. What kinds of eigenvalue bounds do you get from the textbook or from the associated notes? – 2017-02-06

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