Question

The values of \( \mu \) which satisfy the equation \( A^{100} \vec{x} = \mu \vec{x} \), where \( A = \begin{bmatrix} 2 & -1 \\ 0 & -2 \\ 1 & 1 \end{bmatrix} \) are

• A. \( 0, 0, 0 \)
• B. \( 1, 1, 1 \)
• C. \( -1, -1, 1 \)
• D. \( 0, 1, 1 \)

Hint:

Eigenvalues of a matrix give us the scaling factor for the corresponding eigenvector in the equation \( A \vec{x} = \mu \vec{x} \).

Answer 1

The Correct Option is D

To solve for \( \mu \), we look for the eigenvalues of the matrix \( A \). After performing the necessary operations to find the eigenvalues, we obtain the values \( 0, 1, 1 \). These values of \( \mu \) satisfy the equation.