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To estimate the eigenvalues of the given matrix using Gerschgorin bounds, let’s go step by step. ### Matrix \(A\): \[ A = \begin{bmatrix} 1 & 2 & -1 \\ 1 & 1 & 1 \\ 1 & 3 & -1 \end{bmatrix} \] ### Gerschgorin Circle Theorem: The Gerschgorin Circle Theorem states that every eigenvalue of \(A\) lies within at least one of the Gerschgorin discs, each centered at \(a_{ii}\) with a radius equal to the sum of the absolute values of the non-diagonal entries in the corresponding row. ### Step 1: Identify Centers and Radii - Row 1: - Center: \(a_{11} = 1\) - Radius: \(|2| + |-1| = 3\) - Gerschgorin Disc: Centered ______ ____ ____ ____ __________ _____ _________.
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