Question

Let \( M \) be a \( 3 \times 3 \) matrix with real entries such that

\[ M^2 = M + 2I, \quad \text{where} \quad I \text{ denotes the } 3 \times 3 \text{ identity matrix.} \]

If \( \alpha, \beta, \gamma \) are eigenvalues of \( M \) such that \( \alpha\beta\gamma = -4, \) then \( \alpha + \beta + \gamma \) is equal to .............

Hint:

To find the sum or product of eigenvalues, use the fact that the eigenvalues of a matrix satisfy the characteristic equation.

Answer 1

Correct Answer: 3

Step 1: Use the given condition \( M^2 = M + 2I \).
The matrix equation \( M^2 = M + 2I \) suggests that the eigenvalues of \( M \) satisfy the equation: \[ \lambda^2 = \lambda + 2, \] where \( \lambda \) represents the eigenvalue of \( M \).
Step 2: Solve for the eigenvalues.
Rearranging the equation: \[ \lambda^2 - \lambda - 2 = 0. \] Factoring the quadratic equation: \[ (\lambda - 2)(\lambda + 1) = 0. \] Thus, the eigenvalues of \( M \) are \( \lambda = 2 \) and \( \lambda = -1 \).
Step 3: Use the relationship between the eigenvalues.
We are given that \( \alpha \beta \gamma = -4 \), and since there are three eigenvalues for \( M \), we conclude that: \[ \alpha = 2, \quad \beta = -1, \quad \gamma = -2. \]
Step 4: Compute the sum of the eigenvalues.
Thus, the sum of the eigenvalues is: \[ \alpha + \beta + \gamma = 2 + (-1) + (-2) = -1. \]
Step 5: Conclusion.
Therefore, \( \alpha + \beta + \gamma \) is: \[ \boxed{-1}. \]