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G5 a (b) 2. Write the adjacency matrices for the following graphs. (a) G₁ (b) G₂ a b c b (c) b (d) G4 (f) Go a 00110 0 0110 1 1 1 1 000 1 1001 1 01 0 1 00 011 10 1 a с 3. Compute the square of the adjacency matrix for the following graphs from Exercise 2. (a) G₁ (b) G₂ (c) G3 (d) G4 (e) G5 (f) Go Use your answer to tell how many paths of length 2 there are in each graph between vertex a and vertex d. Section 2.4 1. Evaluate the following products involving the 1's vector 1, the identity matrix I, and the ith unit vector e, = [0, 0,...,1..., 0, 0]. Assume that all have size n. (a) I² (f) e, e (b) 1.1 (c) I1 (g) e, e, (ij) (h) 111 . (d) Ie, (e) 1 ·e; (i) e,le, (i = j) 2. (a) Write the following system of equations in matrix form. 3x₁ + 5x₂ + 7x3 = 8 2x₁ x₂ + x3 = 4 2x3 = 6 - x₁ + 6x₂ = (b) Rewrite the matrix equation in part (a) to reflect the operation of bringing the right side over to the left side (so that the right sides are now 0's). 6. (a) Consider the following system of equations for the growth of rabbits and foxes from year to year: R' = 1.5R .2F + 100 F' = 3R + 9F + 50 = Write this system in matrix form, where p [R', F']. (b) Write a matrix equation for p", the vector of rabbits and foxes after 2 years. (c) Write a matrix equation for p(3), the vector of rabbits and foxes after 3 years. [R, F] and p' (d) Using summation notation (E), write a matrix equation for p"), the vector of rabbits and foxes after n years. 10. Let A and B be 2-by-2 matrices and x, y, z be 2-vectors such that Ax = By = [1, 1], Ay = [1, 0], Bx = [0, 1]. Determine z when (a) z = A(2x - y) (b) z = (A - B)x (A + B)x - 2(A + B)y (d) z (c) z = (e) z = [(A + B)y] [(A + 3B)(x - y)]1 = (3A + B)(x + y) Section 2.6 3. Partition the following matrices into appropriate submatrices. [2 2 2 2 1 1 1 2 2 2 11 1 2 1 2 2 22 1 1 1 1 1 1 2 2 2 2 222 2 21111 2 2 2 2 111 1 11112222 1 1 2 2 2 2 (a) (b) 1 010 0101 0101 101 0 10 1 0010 1 01 011010 20200101 02 021010 20 2001 01 0 2 0 2 1 0 1 0 Determine the square of the matrix in Exercise 3, part (b). 6. Write the adjacency matrices of the following graphs and define a par- titioned form of the matrices. (a) G₁ b f (b) G₂ a b C g d 7. Consider the following Markov chain model. B C A D H E F G In the maze, a person in any given room has equal chances of leaving by any door out of the room (but never remains in a room). Write the Markov transition matrix for this maze. Write the matrix in partitioned form.