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Eigenvalues : For an n x n matrix A, show that if one or more of the eigenvalues is zero, A has no inverse. Also show that if, A does have an inverse, the eigenvalues of A^-1 are the reciprocals...

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For an n x n matrix A, show that if one or more of the eigenvalues is zero, A has no inverse.

Also show that if, A does have an inverse, the eigenvalues of A^-1 are the reciprocals of the eigenvalue A.

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Solution Summary

Eigenvalue proofs are provided. The solution is detailed and well presented.

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Proof:
1. We know, if A has an inverse, then the equation Ax=b has a unique solution x=A^(-1)*b.
But from the condition, at least one of the eigenvalue of A is 0. By the definition of ...

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