MATH 2050 Linear Algebra

(Section 4)
2009 Winter

Assignment 4   -   Questions

[Sections 2.1 - 2.3   Matrix Algebra;
linear systems - basic solutions;
block matrices;   adjacency matrices;
inverse matrices (2×2, by block, by Gaussian elimination)]
Due in class on 2009 February 09 (Monday)

  1. Find the basic solutions of the homogeneous linear system AX = O, where         [15 marks]
    A = [ 1 2 -3 4 0 8 ] [ 2 4 -6 9 2 16 ] [ 0 0 0 3 6 1 ] [ 0 0 0 1 2 0 ]
    Show that   Xp = [ 5 0 0 1 0 -1 ]T   is a solution to the inhomogeneous system   AX = B, where   B = [ 1 3 2 1 ]T.
    Hence write down the general solution of   AX = B.

  1. Find   A -1   and hence solve   AX = B, where   A = and B = [ 10 -6 ] [ 6 ] [ 3 -2 ] [ 2 ]         [10 marks]

  1. Use block multiplication to find   AB , where                 [10 marks]
    A = and B = [ 2 5 0 0 0 ] [ 8 3 0 0 0 ] [ 1 2 0 0 0 ] [ 3 1 0 0 0 ] [ 0 0 1 0 0 ] [ 0 0 3 0 0 ] [ 0 0 0 1 0 ] [ 0 0 0 2 0 ] [ 0 0 0 0 1 ] [ 0 0 0 0 1 ]

  1. Use block multiplication to find   C -1, where                 [10 marks]
    C = [ 2 5 0 0 ] [ 1 2 0 0 ] [ 0 0 3 -8 ] [ 0 0 -1 3 ]

  1. Directed Graphs   (Textbook, page 46)         [25 marks]

    Adam Ant is visiting his four aunts at their homes that are connected by twigs as shown in this directed graph:
    [D <---> A <---> B <---> C and A -> C and C -> C]

    There is only one twig connecting homes A and B in each direction.
    Similarly, there is one twig connecting homes B and C in each direction.
    There is one twig connecting homes D and A in each direction.
    However, Adam can travel along the other two twigs only in the directions indicated.
    In the terminology of directed graphs,
    each aunt’s home is a vertex and each twig is an edge.
    We can set up an adjacency matrix A to represent this information.

    If there is an edge from vertex j to vertex i, then the entry aij
    (in the ith row and jth column of the matrix A) is a '1'.
    If there is no edge, then the entry is a zero.

    Write down the adjacency matrix   A   for Adam Ant.

    Furthermore, an r-path (or path of length r) from vertex j to vertex i is a sequence of r edges that starts at vertex j and ends at vertex i.   The 2-paths for Adam that start at home A and end at home C are:
    From: via:   to:  
    A C C
    A B C
    There are two 2-paths from A to C, so that the entry in row 3, column 1 of the associated matrix is '2'.
    Extend this table to find all 2-paths and hence show that the adjacency matrix for 2-paths is
            [ 2 0 1 0 ] [ 1 2 1 1 ] [ 2 2 2 1 ] [ 0 1 0 1 ]
    Also show that this matrix is the square of the matrix A.

    In general the adjacency matrix for r-paths is just Ar.
    Find the number of 3-paths from home C to itself and list all such paths.

  1. Use Gaussian elimination to find the inverse matrix of         [10 marks]
    F = [ 1 -4 -6 8 ] [ 0 2 4 6 ] [ 0 0 -2 4 ] [ 0 0 0 -1 ]
    (if possible; otherwise to show that no such inverse exists).

  1. Use Gaussian elimination to find the inverse matrix of         [10 marks]
    G = [ 1 2 3 4 ] [ 2 5 6 8 ] [ 6 7 8 9 ] [ 4 3 2 1 ]
    (if possible; otherwise to show that no such inverse exists).

  1. Use Gaussian elimination to find the inverse matrix of         [10 marks]
    H = [ 0 1 1 ] [ 1 0 1 ] [ 1 1 0 ]
    (if possible; otherwise to show that no such inverse exists).

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      Created 2009 01 06 and most recently modified 2009 01 28 by Dr. G.H. George