I've identified (a). It's (b)—(g) that I'd really appreciate help with.
Consider the graph U2 (a) Find the adjacency matrix A- A(G) (b) Compute A4 and useit to determine...
Solve all parts please 5. In the following problems, recall that the adjacency matrix (or incidence matrix) for a simple graph with n vertices is an n x n matrix with entries that are all 0 or 1. The entries on the diagonal are all 0, and the entry in the ih row and jth column is 1 if there is an edge between vertex i and vertex j and is 0 if there is not an edge between vertex...
Question 1: Given the following matrix A. 02 A- 1 2 3 2 (a) Find the determinant of A (b) Find eigenvalues and the corresponding eigenspaces of A (c) Determine whether A is diagonalizable. If so, find a matrix P and a diagonal matrix D such that P-1AP=D If not, justify your answer. (d) Find a basis of Im(A) and find the rank of Im(A) (e) Find a basis of Ker(A) and find the rank of Ker(A) Question 1: Given...
69. 2* Use reasoning similar to the previous exercise to find the eigenvalues of the adja- cency matrix of the complete bipartite graph Krs. Thus first compute the number of closed walks of length l in Krs. 69. 2* Use reasoning similar to the previous exercise to find the eigenvalues of the adja- cency matrix of the complete bipartite graph Krs. Thus first compute the number of closed walks of length l in Krs.
7. Graphs u, u2, u3, u4, u5, u6} and the (a) Consider the undirected graph G (V, E), with vertex set V set of edges E ((ul,u2), (u2,u3), (u3, u4), (u4, u5), (u5, u6). (u6, ul)} i. Draw a graphical representation of G. ii. Write the adjacency matrix of the graph G ii. Is the graph G isomorphic to any member of K, C, Wn or Q? Justify your answer. a. (1 Mark) (2 Marks) (2 Marks) b. Consider an...
(a) Find the eigenvalues of the matrix 4) 2 1' and find an eigenvector corresponding to each eigenvalue. Hence find an invertible matrix, P, and a diagonal matrix, D, such that P-1AP = D. (b) Use your result from (a) to find the functions f(t) and g(t) such that f(t)-f(t) +2g(t) g(t) 2f(t) g(t), where f(0)-1 and g(0)-2 (c) Now suppose that f(0)-α and g(0) β. Determine the condition(s) on α and β that must hold if, as t,t is...
1. Consider the matrix (a) Find the characteristic polynomial and eigenvalues of A (b) Find a basis for the eigenspace corresponding to each eigenvalue of A. (c) Find a diagonalization of A. That is, find an invertible matrix P and a diagonal matrix such that A - POP! (d) Use your diagonalization of A to compute A'. Simplify your answer.
8, (10 pts) Show that given a directed graph G = (V,E) already stored in adjacency matrix form, determining if there is a vertex with in-degree n - 1 and out-degree 0 can be done in O(n) time where n is the number of vertices in V. 8, (10 pts) Show that given a directed graph G = (V,E) already stored in adjacency matrix form, determining if there is a vertex with in-degree n - 1 and out-degree 0 can...
4. Consider the following matrix [1 0 -27 A=000 L-2 0 4] (a) (3 points) Find the characteristic polynomial of A. (b) (4 points) Find the eigenvalues of A. Give the algebraic multiplicity of each eigenvalue (c) (8 points) Find the eigenvectors corresponding to the eigenvalues found in part (b). (d) (4 points) Give a diagonal matrix D and an invertible matrix P such that A = PDP-1 (e) (6 points) Compute P-and verify that A= PDP- (show your steps).
Find the eigenvalues of the given matrix. [-14 -6 36 16 1) A) -2.-4 B)-4 C)-2 D) -24 The characteristic polynomial of a 5 5 matrix is given below. Find the eigenvalues and their multiplicities 2) A5 - 24A4-189A3-486A2 2) A) 0 (multiplicity 2),-9 (multiplicity 2),-6 (multiplicity 1) B) 0 (multiplicity 1),9 (multiplicity 3), 6 (multiplicity ) C) 0 (multiplicity 2),9 (multiplicity 2),6 (multiplicity 1) D) 0 (multiplicity 2),-9 (multiplicity 2),6 (multiplicity 1) Diagonalize A- PDP-1 the matrix A, if...
Can you please solve this fully Question 9 (10 marks) (i) How many vertices and how many edges do each of the following graphs have? [3 marks] (b) C16 (a) K70 (d) K2,5 (ii Suppose you have a graph G with vertices vi, v. vi7. Explain (clearly) how you would use the adjacency matrix A to find a. The number of paths from v to vir of length 12.12 marks] b. The length of a shortest path from vi to...