3. Consider the directed acyclic graph (DAG) shown below. (a) Write down the adjacency matrix A corresponding to the ordering of the vertices given by alphabetical order: a, b, c, d, e, ƒ. (b) For the matrix A found in part (a), compute the matrix (I – A)-1, where I denotes the 6 x 6 identity matrix, and show all your work. (Hint: it is much easier to do this by counting paths, rather than using linear algebra! But if you insist on doing this using linear algebra, you must still show all your work.) (c) Find all permutations of the set of vertices {a, b, c, d, e, ƒ} such that the associ- ated adjacency matrix is strictly lower triangular.
3. Consider the directed acyclic graph (DAG) shown below. (a) Write down the adjacency matrix A corresponding to the ordering of the vertices given by alphabetical order: a, b, c, d, e, ƒ. (b) For the matrix A found in part (a), compute the matrix (I – A)-1, where I denotes the 6 x 6 identity matrix, and show all your work. (Hint: it is much easier to do this by counting paths, rather than using linear algebra! But if you insist on doing this using linear algebra, you must still show all your work.) (c) Find all permutations of the set of vertices {a, b, c, d, e, ƒ} such that the associ- ated adjacency matrix is strictly lower triangular.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:**Graph Description:**
The image shows a directed acyclic graph (DAG) with the following vertices: a, b, c, d, e, f. The directed edges indicated by arrows are:
- a → d
- c → a
- c → b
- b → d
- d → f
- e → b
- e → f
**Tasks:**
(a) **Adjacency Matrix:**
- Write down the adjacency matrix \( A \) for the vertices ordered alphabetically as a, b, c, d, e, f.
(b) **Matrix Computation:**
- For matrix \( A \) from part (a), compute \( (I - A)^{-1} \), where \( I \) is the \( 6 \times 6 \) identity matrix. It's suggested to do this by counting paths rather than relying only on linear algebra, but all work must be shown if linear algebra is used.
(c) **Permutations for Strictly Lower Triangular Matrix:**
- Find all permutations of the vertex set \(\{a, b, c, d, e, f\}\) so that the adjacency matrix becomes strictly lower triangular.
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