# Matrices, Reciprocals and Characteristic Vectors

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If A is nonsingular, show that the characteristic values of A^(-1) are the reciprocals of A, and that A and A^(-1) have the same characteristic vectors.

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#### Solution Preview

Proof. Since A is nonsingular, the determinant of A is not zero. So, zero is not an eigenvalue of A. ...

#### Solution Summary

It is shown that if A is nonsingular, that the characteristic values of A^(-1) are the reciprocals of A, and that A and A^(-1) have the same characteristic vectors.

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