1
0
1
0
​
  
0
0
0
0
​
  
1
0
1
0
​
  
0
0
0
1
​
  
is this matrix transitive

To determine whether the given matrix is transitive, we need to verify if it satisfies the **transitivity property** of a relation.

### Transitivity Property:
A relation \( R \) on a set is **transitive** if, whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).

The given matrix is:
\[
\begin{pmatrix}
1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}
\]

This is a **4x4** matrix, which represents a relation on a set of 4 elements. The entry in row \( i \) and column \( j \) is 1 if there is a relation between element \( i \) and element \( j \), and 0 otherwise.

### Steps to check for transitivity:

1. For each pair \( (i, j) \) where \( M[i][j] = 1 \), check if for all \( k \), whenever \( M[j][k] = 1 \), we have \( M[i][k] = 1 \) as well.

- **Row 1**: \( (1, 1) = 1 \), \( (1, 3) = 1 \)
  - Check if these pairs are transitive:
    - For \( (1, 1) \), check all possible pairs with row 1. Since row 1 does not have further links, it's fine.
    - For \( (1, 3) \), check if \( M[3][k] \) holds transitivity. Since \( M[3][3] = 1 \), we must check if \( M[1][3] = 1 \) (which is true), so it holds.

- **Row 2**: All entries are zero, so no transitivity violations here.

- **Row 3**: \( (3, 1) = 1 \), \( (3, 3) = 1 \)
  - Check:
    - For \( (3, 1) \), since row 1 does not have further relations, it's fine.
    - For \( (3, 3) \), check transitivity. Similar to row 1, no violations.

- **Row 4**: \( (4, 4) = 1 \), no further links, so no issues.

### Conclusion:
Since no violations of the transitivity property were found, **the given matrix is transitive**.

Would you like more details or have any questions?

Here are some related questions for deeper understanding:
1. What are other properties like reflexivity and symmetry in relations?
2. How do we check if a matrix represents a reflexive relation?
3. How would the matrix change if it were not transitive?
4. What are some examples of non-transitive relations?
5. How is matrix multiplication related to transitivity in relations?

**Tip**: When checking for transitivity in matrices, think of multiplying the matrix with itself to see if the relation holds for paths longer than one step.

Learn how to determine if a given 4x4 matrix satisfies the transitivity property of a relation. This problem involves matrix theory and the transitivity theorem. Suitable for high school students in Grades 9-12.

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