This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Practice problems for Midterm 2 1. Suppose that A is a 100 × 100 matrix with all elements equal to zero except for the diagonal elements, a ii = 3. (a) Calculate  A  .  A  = 3 100 , because B is a diagonal matrix. (b) Suppose that the matrix B is the same a A , except is has one more nonzero element: b 13 , 74 = 5. Calculate  B  .  B  = 3 100 , because B is an upper triangular matrix. (c) Suppose that the matrix C is the same a B , except is has one more nonzero element: c 81 , 80 = 7. Calculate  C  . Let us expand C in terms of the cofactors of row 81. This expansion contains only two terms,  C  = c 81 , 81 ( 1) 81+81 M 81 , 81 + c 81 , 80 ( 1) 81+80 M 81 , 80 . We notice that the first term can be evaluated: M 81 , 81 is the deter minant of an upper triangular matrix (by crossing out the 81th row, we get rid of the nonzero element in the lower part of the matrix). Therefore, M 81 , 81 = 3 99 . On the other hand, M 81 , 80 = 0 because this is the determinant of a matrix which has a row (the 80th row) consisting entirely of zeros. Therefore, we have,  C  = 3 100 . 2. Suppose that the matrix A is given by A = 1 4 3 1 11 16 9 11 2 1 2 1 10 9 1 . (a) Does the system AX = 0 have nonzero solutions? Why? Yes. We find the reduced form of A among the “Useful reduction facts”, (d). We can see that Rank A = 2, and the size of A is n × m , with n = m = 4, so m > Rank A , and thus the homogeneous equation must have nonzero solutions. 1 (b) How many independent variables does the system AX = 0 have? How many dependent variables? Find all the solutions of the system AX = 0. m Rank A = 2 independent variables, and 2 dependent variables. We have x 1 = 3 / 7 x 3 x 4 , x 2 = 6 / 7 x 3 . (c) Suppose we have a nonhomogeneous system AX = B , where B = b 1 b 2 b 3 b 4 . Specify the conditions on the components of B which would guarantee that the equation AX = B has nonzero solutions. In order for the equation AX = B to have solutions, we need to have Rank A = Rank( A  B ). Thus the last two entries of the reduced right hand side must be zero. This gives the following conditions: 3 b 1 2 b 3 b 4 = 0 , b 2 6 b 3 b 4 = 0 ....
View
Full
Document
This note was uploaded on 01/12/2010 for the course MATH 421 taught by Professor Nataliakomarova during the Winter '07 term at San Jose State University .
 Winter '07
 nataliakomarova

Click to edit the document details