Let M be a `3xx3` matrix satisfying `M^(3)=0`. Then which of the following statement(s) are true: (a) `|M^(2)+M+I|ne0` (b) `|M^(2)-M+I|=0` (c) `|M^(2)+M+I|=0` (d) `|M^(2)-M+I|ne0`

Options

A. `|M^(2)+M+I|ne0`
B. `|M^(2)-M+I|=0`
C. `|M^(2)+M+I|=0`
D. `|M^(2)-M+I|ne0`

Answer

To solve the problem, we need to analyze the properties of the matrix \( M \) given that \( M^3 = 0 \). This means that \( M \) is a nilpotent matrix, and we will use this property to evaluate the determinants of the expressions given in the options.

### Step-by-step Solution:

1. **Understanding Nilpotent Matrices**:
Since \( M^3 = 0 \), it implies that \( M \) is a nilpotent matrix. For a \( 3 \times 3 \) nilpotent matrix, all eigenvalues are zero.

2. **Characteristic Polynomial**:
The characteristic polynomial of a \( 3 \times 3 \) nilpotent matrix \( M \) can be expressed as \( \lambda^3 = 0 \). This means that the only eigenvalue is \( 0 \).

3. **Determinants of Expressions**:
We will evaluate the determinants of the expressions given in the options.

- **Option (a)**: \( |M^2 + M + I| \)
- Since \( M \) is nilpotent, \( M^2 \) and \( M \) will also have eigenvalues of \( 0 \). Thus, the eigenvalues of \( M^2 + M + I \) will be \( 0 + 0 + 1 = 1 \) (for each eigenvalue).
- Therefore, \( |M^2 + M + I| = 1^3 = 1 \neq 0 \).

- **Option (b)**: \( |M^2 - M + I| \)
- The eigenvalues of \( M^2 - M + I \) will be \( 0 - 0 + 1 = 1 \) (for each eigenvalue).
- Therefore, \( |M^2 - M + I| = 1^3 = 1 \neq 0 \).

- **Option (c)**: \( |M^2 + M + I| \)
- This is the same as option (a), which we already evaluated as \( 1 \neq 0 \).

- **Option (d)**: \( |M^2 - M + I| \)
- This is the same as option (b), which we already evaluated as \( 1 \neq 0 \).

4. **Conclusion**:
From our evaluations:
- Option (a): True
- Option (b): True
- Option (c): True
- Option (d): True

### Final Answer:
All options (a), (b), (c), and (d) are true.