1 16. Let the adjacency matrix for a relation R on the set {a, b, c) be given by 0 1 1 0 is the composition of R with itself, which ordered pair belongs to R²? O (b,c) O (c,b) O (b,a) 1 none of these 0 0. If R² 0
1 16. Let the adjacency matrix for a relation R on the set {a, b, c) be given by 0 1 1 0 is the composition of R with itself, which ordered pair belongs to R²? O (b,c) O (c,b) O (b,a) 1 none of these 0 0. If R² 0
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![**Matrix Relations and Composition**
**Problem 16:**
Consider the adjacency matrix for a relation \( R \) on the set \(\{a, b, c\}\), given by:
\[
\begin{bmatrix}
1 & 1 & 0 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{bmatrix}
\]
If \( R^2 \) is the composition of \( R \) with itself, which ordered pair belongs to \( R^2 \)?
- (b, c)
- (c, b)
- (b, a)
- none of these
**Explanation:**
An adjacency matrix represents a relation on a set, where each element of the matrix indicates whether a pair of elements in the set is related. The entry at the \(i\)-th row and \(j\)-th column is \(1\) if the element corresponding to the \(i\)-th row is related to the element corresponding to the \(j\)-th column, and \(0\) otherwise.
The composition \( R^2 \) involves matrix multiplication of \( R \) with itself. By performing this operation, we can determine which new pairs in the set are related under the composition. The goal is to identify the correct ordered pair from the options provided that appears in \( R^2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9348d578-66ea-4064-b97e-301aeae52239%2Ffa0921af-314b-481c-b483-dbfc2d9a39a4%2Fa2x2wu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Matrix Relations and Composition**
**Problem 16:**
Consider the adjacency matrix for a relation \( R \) on the set \(\{a, b, c\}\), given by:
\[
\begin{bmatrix}
1 & 1 & 0 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{bmatrix}
\]
If \( R^2 \) is the composition of \( R \) with itself, which ordered pair belongs to \( R^2 \)?
- (b, c)
- (c, b)
- (b, a)
- none of these
**Explanation:**
An adjacency matrix represents a relation on a set, where each element of the matrix indicates whether a pair of elements in the set is related. The entry at the \(i\)-th row and \(j\)-th column is \(1\) if the element corresponding to the \(i\)-th row is related to the element corresponding to the \(j\)-th column, and \(0\) otherwise.
The composition \( R^2 \) involves matrix multiplication of \( R \) with itself. By performing this operation, we can determine which new pairs in the set are related under the composition. The goal is to identify the correct ordered pair from the options provided that appears in \( R^2 \).
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