Question

What is the dimension of the vector space of all \( n \times n \) real symmetric matrices?

• A. \( n^2 \)
• B. \( \frac{n(n+1)}{2} \)
• C. \( \frac{n(n-1)}{2} \)
• D. \( 2n \)

Hint:

Symmetric matrix dimension = diagonal + upper triangle
= \( n + \frac{n(n-1)}{2} = \frac{n(n+1)}{2} \)

Answer 1

The Correct Option is B

Concept: A real symmetric matrix satisfies: \[ A^T = A \] This means the entries are symmetric about the main diagonal.
Step 1: Count diagonal elements
There are \( n \) diagonal elements: \[ a_{11}, a_{22}, \dots, a_{nn} \]
Step 2: Count off-diagonal elements
For \( i \neq j \), we have: \[ a_{ij} = a_{ji} \] So each pair contributes only one independent element. Number of such pairs: \[ \frac{n(n-1)}{2} \]
Step 3: Total independent elements
\[ n + \frac{n(n-1)}{2} = \frac{n(n+1)}{2} \] Hence, the dimension is: \[ \frac{n(n+1)}{2} \]