mid2practsol

# mid2practsol - Practice problems for Midterm 2 1 Suppose...

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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 co-factors 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 non-zero 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 non-zero 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 non-homogeneous 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 non-zero 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 ....
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mid2practsol - Practice problems for Midterm 2 1 Suppose...

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