Question

For any real symmetric matrix \( A \), the transpose of \( A \) is ________ .

• A. inverse of \( A \)
• B. null matrix
• C. \( -A \)
• D. \( A \)

Hint:

For symmetric matrices, always remember that the matrix is equal to its transpose: \( A = A^T \). This property is fundamental in linear algebra and is used in various matrix-related computations.

Answer 1

The Correct Option is D

To solve this, we need to recall the definition of a symmetric matrix. A matrix \( A \) is said to be symmetric if it is equal to its own transpose, i.e.,
\[ A = A^T \]
Step 1: Definition of Transpose
The transpose of a matrix \( A \), denoted by \( A^T \), is obtained by swapping its rows and columns. So, for any matrix \( A = [a_{ij}] \), the transpose matrix \( A^T \) will have elements \( A^T = [a_{ji}] \).

Step 2: Symmetry Condition
For a symmetric matrix, by definition, the matrix is equal to its transpose. This implies:
\[ A = A^T \]
In other words, the elements of the matrix satisfy the condition that \( a_{ij} = a_{ji} \) for all \( i, j \).

Step 3: Conclusion
Since a real symmetric matrix satisfies \( A = A^T \), the transpose of \( A \) is exactly \( A \). Therefore, the correct answer is \( A \).

\[ A^T = A \]