The matrix `[(1,0,0),(0,2,0),(0,0,4)]` is a

Options

A. identity matrix
B. symmetric matrix
C. skew - symmetric matrix
D. None of these

Answer

To determine the type of the matrix \( A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{pmatrix} \), we will analyze it step by step.

### Step 1: Identify the Matrix
The given matrix is:
\[ A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{pmatrix} \]

### Step 2: Check for Identity Matrix
An identity matrix is a square matrix where all the diagonal elements are 1 and all other elements are 0.

For matrix \( A \):
- The diagonal elements are 1, 2, and 4.
- Since not all diagonal elements are 1, \( A \) is **not** an identity matrix.

### Step 3: Check for Symmetric Matrix
A matrix is symmetric if \( A = A^T \), where \( A^T \) is the transpose of \( A \).

To find the transpose \( A^T \):
\[ A^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{pmatrix} \]

Now, we compare \( A \) and \( A^T \):
\[ A = A^T \]
Since both matrices are equal, \( A \) is a **symmetric matrix**.

### Step 4: Check for Skew-Symmetric Matrix
A matrix is skew-symmetric if \( A = -A^T \).

From the previous step, we know:
\[ A^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{pmatrix} \]
Thus, \( -A^T = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -4 \end{pmatrix} \).

Since \( A \neq -A^T \), \( A \) is **not** a skew-symmetric matrix.

### Conclusion
Based on the analysis, the matrix \( A \) is a **symmetric matrix**.

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