The matrix `[(1,0,0),(0,2,0),(0,0,0)]` is a :

Options

A. Identity matrix
B. Symmetric matrix
C. Skew symmetric matrix
D. None of these

Answer

To determine the type of the given matrix \( A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{pmatrix} \), we will analyze its properties step by step.

### Step 1: Identify the Matrix
The given matrix is:
\[ A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]

### Step 2: Check for Symmetry
A matrix is symmetric if \( A^T = A \), where \( A^T \) is the transpose of matrix \( A \).

**Finding the Transpose:**
To find the transpose of matrix \( A \), we swap its rows and columns:
\[ A^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]

**Comparison:**
Now, we compare \( A^T \) with \( A \):
\[ A^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{pmatrix} = A \]
Since \( A^T = A \), the matrix is symmetric.

### Step 3: Check for Skew-Symmetry
A matrix is skew-symmetric if \( A^T = -A \).

**Finding -A:**
\[ -A = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]

**Comparison:**
Now, we compare \( A^T \) with \(-A\):
\[ A^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{pmatrix} \neq -A = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]
Since \( A^T \neq -A \), the matrix is not skew-symmetric.

### Conclusion
Since the matrix is symmetric and not skew-symmetric, we conclude that the given matrix is a symmetric matrix.

### Final Answer
The matrix \( A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{pmatrix} \) is a symmetric matrix.

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