CS 3333 Mathematical Foundations Spring '12
Solve the following problems from the textbook [KR]. For each problem, show the intermediate steps and the final answer. Matrices: Pages 183-185: 1 (use matrix A in problem 2a to solve this problem), 2b, 3 b & c, 6 (solve using matrix inverses), 8, 9, 10, 12a, 14, 15 (calculate a few powers of A and then generalize it to a formula), 16, 17, 18 (calculate both ways: first matrix * second matrix; second matrix*first matrix), 20, 21 (indicate the prior results used at each step), 22, 23, 25 (give the matrix equation-vector equation Ax=B and calculate A-1 and A-1B). Extra question 1: What is the most efficient way to multiply the matrices A1, A2, A3 and A4 if the dimensions of these matrices are 10x10, 10x5, 5x20, and 20x3, respectively? (Give the number of multiplications required for each of the five approaches.) Extra question 2: Compute the determinants of the matrix A in problems 2a and 3a. Page 188: 37, 38, 39. Extra question 3: From the definition of the matrix product, devise an
algorithm for computing the product of two lower triangular matrices that
ignores those products in the computation that are automatically equal to zero. How many multiplications of entries are used by this algorithm
to multiply two nxn upper triangular matrices? |
Rajendra V. Boppana |