Problem 7. For the directed graph below, find the adjacency matrix A, compute A3 , and determine the number of paths of length 3 (3-paths) from v1 to v4 and from v2 to v3. v1 v2 v4 v3

Problem 8. Let A = 1 2 1 0 . Find the matrix B = x y z w such that AB = 1 0 0 1 .

Problem 9. Show that the matrix a b 5a 5b is not invertible, i.e., does not have an inverse.

Problem 10. Find the determinant, the adjugate, and the inverse of the matrix A = 1 1 7 11

Problem 11. Using the matrix inverse, solve the following system of linear equations 5x 6y = 1 x + 11y = 4