Let A be a symmetric matrix, then `A^(m), m in N` is a

Options

A. Singular matrix
B. Null matrix
C. Symmetric matrix
D. Unit matrix

Answer

To show that if \( A \) is a symmetric matrix, then \( A^m \) (where \( m \) is a natural number) is also a symmetric matrix, we can follow these steps:

### Step 1: Definition of Symmetric Matrix
A matrix \( A \) is symmetric if \( A = A^T \), where \( A^T \) denotes the transpose of matrix \( A \).

### Step 2: Expressing \( A^m \)
We want to find \( A^m \) for a natural number \( m \). We can express \( A^m \) as:
\[ A^m = A \cdot A \cdot A \cdots A \quad (m \text{ times}) \]

### Step 3: Taking the Transpose of \( A^m \)
Now, we will take the transpose of \( A^m \):
\[ (A^m)^T = (A \cdot A \cdots A)^T \]
Using the property of transposes, we know that the transpose of a product of matrices is the product of their transposes in reverse order:
\[ (A^m)^T = A^T \cdot A^T \cdots A^T \quad (m \text{ times}) \]

### Step 4: Substituting the Symmetric Property
Since \( A \) is symmetric, we have \( A^T = A \). Therefore, we can substitute \( A \) for \( A^T \):
\[ (A^m)^T = A \cdot A \cdots A \quad (m \text{ times}) = A^m \]

### Step 5: Conclusion
Since \( (A^m)^T = A^m \), we conclude that \( A^m \) is also symmetric.

Thus, the final result is:
\[ A^m \text{ is symmetric for all } m \in \mathbb{N}. \]