If a is an eigenvalue of a square matrix M, then 1/a is an eigenvalue of

If \( \lambda \) is an eigenvalue of a square matrix \( M \), then \( \frac{1}{\lambda} \) is an eigenvalue of the inverse matrix \( M^{-1} \), provided that \( M \) is invertible and \( \lambda \neq 0 \).

### Explanation:
1. By definition, if \( \lambda \) is an eigenvalue of the matrix \( M \), then there exists a non-zero vector \( v \) (an eigenvector) such that:

\[
M v = \lambda v
\]

2. If \( M \) is invertible, multiplying both sides of the equation by \( M^{-1} \) gives:

\[
M^{-1} M v = M^{-1} \lambda v
\]

Simplifying this:

\[
v = \lambda M^{-1} v
\]

Dividing both sides by \( \lambda \) gives:

\[
M^{-1} v = \frac{1}{\lambda} v
\]

Thus, \( \frac{1}{\lambda} \) is an eigenvalue of \( M^{-1} \).

### Summary:
If \( \lambda \) is an eigenvalue of a square matrix \( M \), then \( \frac{1}{\lambda} \) is an eigenvalue of the inverse matrix \( M^{-1} \).

Do you want more details on eigenvalues and eigenvectors, or have any questions?

Here are 5 related questions for further exploration:
1. What happens to the eigenvalues of a matrix when it is transposed?
2. How do eigenvalues relate to the determinant of a matrix?
3. What are the eigenvalues of a diagonal matrix?
4. How do eigenvalues change under matrix multiplication?
5. How can eigenvectors be used in applications like principal component analysis (PCA)?

**Tip:** Eigenvalues of symmetric matrices are always real.

Explore how eigenvalues of a matrix M relate to the eigenvalues of its inverse matrix M^{-1}. Learn the mathematical proofs and applications of these concepts in linear algebra. Suitable for advanced undergraduate students and researchers.

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