Question

question 10

1 pts

a and its inverse have the same eigenvalues.

true
false

Explanation:

Define eigenvalue relation

Let \(\lambda\) be an eigenvalue of an invertible matrix \(A\).
This means there exists a non-zero vector \(v\) such that:

$$Av = \lambda v$$

Relate to the inverse matrix

Since \(A\) is invertible, \(\lambda
eq 0\).
Multiply both sides of the equation by \(A^{-1}\):

$$A^{-1}(Av) = A^{-1}(\lambda v)$$

$$v = \lambda (A^{-1}v)$$

Solve for the inverse eigenvalue

Divide both sides by the non-zero scalar \(\lambda\):

$$A^{-1}v = \frac{1}{\lambda} v$$

Thus, the eigenvalues of \(A^{-1}\) are the reciprocals \(\frac{1}{\lambda}\).

Analyze the statement

The eigenvalues of \(A\) and \(A^{-1}\) are generally different.
They are only the same if \(\lambda = \frac{1}{\lambda}\) for all eigenvalues.
This is not true for all invertible matrices.
Therefore, the statement is false.

Answer:

• True
• (B) False (Correct answer)