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: 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}$$
2. Matrix 2: 0 & -2 & 3 \\ 2 & 0 & 4 \\ 3 & 4 & 0 \end{pmatrix}$$ 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: 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}$$
4. Matrix 4: 2 & 1 & -2 \\ 1 & 1 & 2 \\ -2 & 2 & 4 \end{pmatrix}$$ 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.