If `A=[(3,0,0),(0,2,0),(0,0,1)]` then A is

Options

A. Diagonal matrix
B. Scalar matrix
C. Nilpotent matrix
D. Idempotent matrix

Answer

To determine the type of matrix \( A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \), we will analyze its properties based on the definitions of the various types of matrices given in the options.

### Step-by-Step Solution:

1. **Identify the Matrix Type**:
- The matrix \( A \) is a square matrix of order 3 (3x3).
- It has non-zero elements only on the diagonal, which are \( 3, 2, \) and \( 1 \), while all off-diagonal elements are zero.

2. **Check if A is a Diagonal Matrix**:
- A diagonal matrix is defined as a matrix where all off-diagonal elements are zero.
- Since all off-diagonal elements of \( A \) are zero, \( A \) is indeed a diagonal matrix.

3. **Check if A is a Scalar Matrix**:
- A scalar matrix is a special case of a diagonal matrix where all the diagonal elements are the same.
- In matrix \( A \), the diagonal elements are \( 3, 2, \) and \( 1 \), which are not all the same. Therefore, \( A \) is not a scalar matrix.

4. **Check if A is a Nilpotent Matrix**:
- A nilpotent matrix is one for which there exists a positive integer \( k \) such that \( A^k = 0 \).
- Let's compute \( A^2 \):
\[ A^2 = A \times A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \times \begin{pmatrix} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 3^2 & 0 & 0 \\ 0 & 2^2 & 0 \\ 0 & 0 & 1^2 \end{pmatrix} = \begin{pmatrix} 9 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
- Since \( A^2 \neq 0 \) and \( A^k \) will never be zero for any positive integer \( k \), \( A \) is not a nilpotent matrix.

5. **Check if A is an Idempotent Matrix**:
- An idempotent matrix is one where \( A^2 = A \).
- We already computed \( A^2 \) and found \( A^2 = \begin{pmatrix} 9 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{pmatrix} \), which is not equal to \( A \). Therefore, \( A \) is not an idempotent matrix.

### Conclusion:
Based on the analysis:
- \( A \) is a diagonal matrix.
- \( A \) is not a scalar matrix.
- \( A \) is not a nilpotent matrix.
- \( A \) is not an idempotent matrix.

Thus, the answer is that \( A \) is a **diagonal matrix**.

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