Question

The eigenvalues of the matrix
\[ \begin{bmatrix} 1 & -2 & 3 \\ -2 & 2 & -4 \\ 3 & -4 & 7 \end{bmatrix} \] are:

• A. \(0, 1, 5 + \sqrt{31}\)
• B. \(0, 1, 5 - \sqrt{31}\)
• C. \(5 + \sqrt{31}, 5 - \sqrt{31}, 1\)
• D. \(5 + \sqrt{31}, 5 - \sqrt{31}, 0\)

Hint:

When solving for eigenvalues, always start by setting up and solving the characteristic equation det(A−λI)=0.

Answer 1

The Correct Option is D

To solve for the eigenvalues of the matrix, we need to calculate the characteristic equation \(\text{det}(A - \lambda I) = 0\), where \(A\) is the given matrix and \(\lambda\) represents the eigenvalues. Solving the resulting cubic equation gives us the eigenvalues as \(5 + \sqrt{31}\), \(5 - \sqrt{31}\), and \(0\).