If `A` is a nilpotent matrix of odd order `r`, then which of the following is true?

Options

A. (a)`l=(l-A)(l^(r-2)A+l^(r-3)A^(2)+ . . . +A^(r-1))`
B. (b)`l=(l+A)(l^(r-1)+l^(r-2)A+l^(r-3)A^(2)+ . . . .+A^(r-1))`
C. (c)`l=(l+A)(l^(r-1)-l^(r-2)A+l^(r-3)A^(2)- . . . .+A^(r-1))`
D. (d)`l=(l-A)(l^(r-1)-l^(r-2)A+l^(r-3)A^(2)- . . . +A^(r-1))`

Answer

To solve the problem, we need to analyze the properties of a nilpotent matrix. A nilpotent matrix \( A \) is defined as a matrix for which there exists a positive integer \( k \) such that \( A^k = 0 \). Given that \( A \) is a nilpotent matrix of odd order \( r \), we will explore the implications of this property.

### Step-by-Step Solution:

1. **Definition of Nilpotent Matrix**:
A matrix \( A \) is said to be nilpotent if there exists a positive integer \( k \) such that \( A^k = 0 \).

**Hint**: Recall that nilpotent matrices have eigenvalues equal to zero.

2. **Order of the Matrix**:
The order of the matrix \( A \) is given as \( r \), which is an odd integer. This means that the size of the matrix \( A \) is \( r \times r \).

**Hint**: The order of a matrix refers to the number of rows (or columns) it has.

3. **Eigenvalues of Nilpotent Matrices**:
Since \( A \) is nilpotent, all its eigenvalues are zero. This can be derived from the fact that if \( A^k = 0 \), then the characteristic polynomial of \( A \) must have all roots equal to zero.

**Hint**: Consider the implications of the characteristic polynomial of a nilpotent matrix.

4. **Trace of the Matrix**:
The trace of a matrix is the sum of its eigenvalues. Since all eigenvalues of \( A \) are zero, the trace of \( A \) is also zero.

**Hint**: Remember that the trace of a matrix is invariant under similarity transformations.

5. **Determinant of the Matrix**:
The determinant of a nilpotent matrix is also zero. This is because the determinant is the product of the eigenvalues, and since all eigenvalues are zero, the determinant must be zero.

**Hint**: Think about how the determinant relates to the eigenvalues of a matrix.

6. **Conclusion**:
Since \( A \) is a nilpotent matrix of odd order \( r \), we can conclude that:
- The trace of \( A \) is zero.
- The determinant of \( A \) is zero.

Therefore, the correct statements regarding the nilpotent matrix \( A \) are that its trace and determinant are both zero.

### Final Answer:
The correct options regarding the nilpotent matrix \( A \) of odd order \( r \) are:
- The trace of \( A \) is zero.
- The determinant of \( A \) is zero.