Question

Suppose that the characteristic equation of \( M \in \mathbb{C}^{3 \times 3} \) is \[ \lambda^3 + \alpha \lambda^2 + \beta \lambda - 1 = 0, \] where \( \alpha, \beta \in \mathbb{C} \) with \( \alpha + \beta \neq 0 \).
Which of the following statements is TRUE?

• A. \( M(I - \beta M) = M^{-1}(M + \alpha I) \)
• B. \( M(I + \beta M) = M^{-1}(M - \alpha I) \)
• C. \( M^{-1}(M^{-1} + \beta I) = M - \alpha I \)
• D. \( M^{-1}(M^{-1} - \beta I) = M + \alpha I \)

Hint:

When working with matrices, always verify equations by multiplying out both sides and simplifying. Pay special attention to the use of inverses and identity matrices.

Answer 1

The Correct Option is D

We are given the characteristic equation of \( M \), and we need to find the correct logical inference. Let's examine each option step by step. Step 1: Understand the given information:
The characteristic equation is: \[ \lambda^3 + \alpha \lambda^2 + \beta \lambda - 1 = 0. \] We can infer that this equation involves powers of \( M \), and the goal is to manipulate the equation to reach the correct statement. Step 2: Evaluate the options:
- Option (A): \( M(I - \beta M) = M^{-1}(M + \alpha I) \). This does not seem to hold because the structure on both sides of the equation does not match when expanded. - Option (B): \( M(I + \beta M) = M^{-1}(M - \alpha I) \). Similarly, this equation does not simplify correctly according to the given equation. - Option (C): \( M^{-1}(M^{-1} + \beta I) = M - \alpha I \). This is incorrect because the manipulation of inverse terms does not lead to the correct conclusion. - Option (D): \( M^{-1}(M^{-1} - \beta I) = M + \alpha I \). This is the correct choice, as the equation can be simplified and verified through algebraic manipulation based on the characteristic equation. Step 3: Conclusion:
The correct statement is Option (D). By performing the necessary operations, we can confirm that the equation holds true.