# midterm1 - tion x 1-2 x 2 x 3 = 1-x 1 x 2-x 3 = x 1 x 2 kx...

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Linear Algebra, Fall 2002 Midterm 1 Nov 01 2002, 13:10 PM - 15:00 PM (1) (10 pts) A = 1 2 1 1 - 1 0 1 1 1 Compute adj A (2) (15 pts) A ∈ M m,n , A ∈ M n,p . Is it true that ( AB ) T = B T A T ? Explain. (3) (20 pts) How many terms are there in the determinant of A ∈ M 5 , 5 ? Does it contain the following terms? If yes, determine the sign of each term. (a) a 12 a 23 a 34 a 45 a 51 (b) a 13 a 25 a 34 a 41 a 52 (c) a 13 a 24 a 31 a 43 a 54 (4) (30 pts) (a) State Cramer’s rule (need not proof). (b) For what value of k is the following a consistent system (ie. have a unique solu-
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Unformatted text preview: tion)? x 1-2 x 2 + x 3 = 1-x 1 + x 2-x 3 = x 1 + x 2 + kx 3 =-1 (c) For those k , use any method to solve (b). (5) (25 pts) A = ± A 11 A 12 A 22 ² where A 11 and A 22 are nonsingular n × n matrices. Determine whether A is nonsigular. If yes, ﬁnd A-1 by a block version of Gauss or Gauss Jordan elemination. 1...
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