If A is a square matrix, then A is symmetric, if

Options

A. `A^(2)=A`
B. `A^(2)=l`
C. `A^(T)=A`
D. `A^(T)=-A`

Answer

To determine when a square matrix \( A \) is symmetric, we follow these steps:

### Step-by-Step Solution:

1. **Definition of Symmetric Matrix**:
A matrix \( A \) is defined as symmetric if it is equal to its transpose, i.e., \( A = A^T \).

2. **Understanding Square Matrices**:
Since \( A \) is a square matrix, it has the same number of rows and columns. This property is crucial because the transpose of a square matrix is also a square matrix.

3. **Expressing the Matrix**:
We can express any square matrix \( A \) as the sum of a symmetric matrix and a skew-symmetric matrix. This means:
\[ A = S + K \]
where \( S \) is symmetric (\( S = S^T \)) and \( K \) is skew-symmetric (\( K = -K^T \)).

4. **Finding the Conditions for Symmetry**:
To find when \( A \) is symmetric, we set:
\[ A = A^T \]
Substituting the expression from step 3, we have:
\[ S + K = S^T + K^T \]
Since \( S = S^T \) and \( K = -K^T \), we can rewrite this as:
\[ S + K = S - K \]

5. **Simplifying the Equation**:
Rearranging gives us:
\[ K + K = 0 \quad \Rightarrow \quad 2K = 0 \quad \Rightarrow \quad K = 0 \]
This means that the skew-symmetric part \( K \) must be zero for \( A \) to be symmetric.

6. **Conclusion**:
Therefore, for a square matrix \( A \) to be symmetric, it must satisfy:
\[ A = A^T \]

### Final Statement:
A square matrix \( A \) is symmetric if and only if \( A = A^T \).

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