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The problem asks to identify which one of the given matrices has similar properties as the matrix \(\mathbf{B} = [b_{ij}]_{m \times m}\) where \(b_{ij} = b_{ji}\), meaning the matrix \(\mathbf{B}\) is symmetric. 

### Matrix Symmetry
A matrix is symmetric if it is equal to its transpose, i.e., \(b_{ij} = b_{ji}\) for all \(i\) and \(j\). This means that the element at position \((i, j)\) is the same as the element at position \((j, i)\).

### Analyzing the Given Matrices

1. **Matrix 1:**
   \[
   \begin{pmatrix}
   0 & -2 & 3 \\
   -2 & 0 & 4 \\
   3 & 4 & 0
   \end{pmatrix}
   \]
   This matrix is symmetric because:
   \[
   \text{Transpose: } \begin{pmatrix}
   0 & -2 & 3 \\
   -2 & 0 & 4 \\
   3 & 4 & 0
   \end{pmatrix} = \begin{pmatrix}
   0 & -2 & 3 \\
   -2 & 0 & 4 \\
   3 & 4 & 0
   \end{pmatrix}
   \]

2. **Matrix 2:**
   \[
   \begin{pmatrix}
   0 & -2 & 3 \\
   2 & 0 & 4 \\
   3 & 4 & 0
   \end{pmatrix}
   \]
   This matrix is not symmetric because:
   \[
   \text{Transpose: } \begin{pmatrix}
   0 & 2 & 3 \\
   -2 & 0 & 4 \\
   3 & 4 & 0
   \end{pmatrix} \neq \begin{pmatrix}
   0 & -2 & 3 \\
   2 & 0 & 4 \\
   3 & 4 & 0
   \end{pmatrix}
   \]

3. **Matrix 3:**
   \[
   \begin{pmatrix}
   1 & 2 & 3 \\
   2 & -1 & -4 \\
   -3 & -4 & 1
   \end{pmatrix}
   \]
   This matrix is not symmetric because:
   \[
   \text{Transpose: } \begin{pmatrix}
   1 & 2 & -3 \\
   2 & -1 & -4 \\
   3 & -4 & 1
   \end{pmatrix} \neq \begin{pmatrix}
   1 & 2 & 3 \\
   2 & -1 & -4 \\
   -3 & -4 & 1
   \end{pmatrix}
   \]

4. **Matrix 4:**
   \[
   \begin{pmatrix}
   2 & 1 & -2 \\
   1 & 1 & 2 \\
   -2 & 2 & 4
   \end{pmatrix}
   \]
   This matrix is symmetric because:
   \[
   \text{Transpose: } \begin{pmatrix}
   2 & 1 & -2 \\
   1 & 1 & 2 \\
   -2 & 2 & 4
   \end{pmatrix} = \begin{pmatrix}
   2 & 1 & -2 \\
   1 & 1 & 2 \\
   -2 & 2 & 4
   \end{pmatrix}
   \]

### Conclusion:
Matrices 1 and 4 are symmetric and therefore have similar properties as matrix \(\mathbf{B}\). Among the options given, both these matrices satisfy the condition \(b_{ij} = b_{ji}\).

Do you have any questions or need further details?

### Related Questions:
1. What is the general condition for a matrix to be symmetric?
2. Can a non-square matrix be symmetric? Why or why not?
3. How does symmetry in a matrix affect its eigenvalues?
4. Can the product of two symmetric matrices be symmetric? Under what conditions?
5. How is the symmetry of a matrix related to positive definiteness?

### Tip:
To quickly check if a matrix is symmetric, compare the upper triangular part of the matrix with its lower triangular part.

Which one of the given matrices has similar properties as the matrix B where b_{ij} = b_{ji}?

Learn how to identify symmetric matrices in linear algebra problems. This problem focuses on checking the symmetry of a matrix by comparing it with its transpose. Suitable for undergraduate-level students.

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