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It is not possible to uniquely determine a matrix given its eigenvalues. For eigenvalues ${1, 2, 3}$, ${A}$ could be a lot of matrices e.g., any upper triangular matrix with ${1, 2, 3}$ as diagonal elements.

If you’re asking for the eigenvalues of the inverse of ${A}$, then it’s simply the reciprocal of the eigenvalues of ${A}$ i.e., ${1, }$ $\frac{1}{2}$ and $\frac{1}{3}$. (Note that ${A}^{-1}$ always exists because all of ${A}$’s eigenvalues are non-zero).

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atulcse asked Jan 13, 2022
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