Question

Let

Then \[ \lim_{n \to \infty} M^n x \]

• A. does not exist
• B.
• C.
• D.

Hint:

For limits of matrix powers, find the eigenvalues and eigenvectors to determine the long-term behavior of the system.

Answer 1

The Correct Option is C

Step 1: Finding the eigenvalues of \( M \). 
We first find the eigenvalues of matrix \( M \). The characteristic equation is: \[ \det(M - \lambda I) = 0. \] For the matrix \( M \), this gives the equation \( (\lambda - 2)^2 = 0 \), so the eigenvalue is \( \lambda = 2 \). 

Step 2: Finding the eigenvector corresponding to \( \lambda = 2 \). 
To find the eigenvector corresponding to \( \lambda = 2 \), we solve: \[ (M - 2I) v = 0. \] This gives the eigenvector

Step 3: Conclusion. 
Since the matrix is diagonalizable and the eigenvalue \( 2 \) dominates as \( n \to \infty \), the limit of \( M^n x \) as \( n \to \infty \) is the eigenvector 
. 
Therefore, the correct answer is (C).