A matrix with m rows and n columns can be represented as A= [aijmxn, 1≤ i ≤m,1≤ j ≤n, i,j∈ N, where are its elements, i is row index and j is column index.
Let B = [bij]mxm be a matrix such that bij biji. Which one of
matrices has similar properties as the matrix B?

Given the problem statement:

1. **Matrix A** is an \( m \times n \) matrix where \( A = [a_{ij}] \) with \( 1 \leq i \leq m \) and \( 1 \leq j \leq n \).
2. **Matrix B** is an \( m \times m \) matrix where each element \( b_{ij} \) satisfies \( b_{ij} = b_{ji} \).

Since \( B \) is symmetric (\( b_{ij} = b_{ji} \)), we are asked to identify which other matrices have similar properties to matrix \( B \).

### Properties of Symmetric Matrices:
1. **Symmetry**: For any matrix \( B = [b_{ij}] \), if \( B \) is symmetric, then \( b_{ij} = b_{ji} \) for all \( i \) and \( j \). This means that the matrix is equal to its transpose, i.e., \( B = B^T \).
2. **Square Matrix**: Symmetric matrices are necessarily square matrices (same number of rows and columns).

### Similar Matrices:
- A matrix that shares the property of symmetry (like matrix \( B \)) must also be a square matrix and must satisfy \( b_{ij} = b_{ji} \).

Thus, a matrix that has similar properties as \( B \) could be:
- **Any other symmetric \( m \times m \) matrix**.

### Possible Answer:
Matrices that have similar properties as matrix \( B \) are symmetric matrices of the same dimension, i.e., any symmetric \( m \times m \) matrix.

### Further Considerations:
If \( B \) is a general \( m \times m \) symmetric matrix, any matrix that is symmetric and of the same dimension would have similar properties. This could include:
- Identity matrix \( I_m \)
- Zero matrix \( O_m \)
- Any other matrix \( C = [c_{ij}]_{m \times m} \) where \( c_{ij} = c_{ji} \).

Would you like more details or have any questions?

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Here are 5 related questions that might interest you:

1. What are the eigenvalues of a symmetric matrix?
2. How do you prove that a symmetric matrix is always diagonalizable?
3. Can a non-square matrix be symmetric?
4. What is the difference between a diagonal matrix and a symmetric matrix?
5. How do symmetric matrices relate to quadratic forms?

**Tip:** Symmetric matrices are crucial in various fields such as physics and engineering, particularly because of their well-behaved eigenvalues and eigenvectors.

A matrix with m rows and n columns can be represented as A= [aij]mxn, 1≤ i ≤m,1≤ j ≤n, i,j∈ N, where are its elements, i is row index and j is column index. Let B = [bij]mxm be a matrix such that bij = bji. Which one of matrices has similar properties as the matrix B?

Explore the properties of symmetric matrices, focusing on the key concept that bij = bji, and learn how to identify matrices with similar properties. This problem is suitable for undergraduate students studying Linear Algebra.

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