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# exam1_sl - M341 Midterm Exam 1 Solution I.(15 points)...

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M341 Midterm Exam 1 Solution I.(15 points) Examples. 1. Give an example of a 3 × 3 matrix A such that A is symmetric and trace( A ) = 0 but A is not diagonal. A possible example: A = 0 1 0 1 0 0 0 0 0 2. Give an example of two square matrices, A and B , that don't commute. A possible example: A = [ 0 1 0 0 ] , B = [ 1 0 0 0 ] Then AB = [ 0 0 0 0 ] , and BA = [ 0 1 0 0 ] Hence AB ̸ = BA and they don't commute. 3. Describe completely every matrix that is both upper triangular and skew-symmetric. Answer: All square zero matrices. Reason: The diagonal entry of a skew-symmetric matric is always 0 . Since the matrix is both skew-symmetric and upper triangular, its o diagonal entries are also zero. Therefore, the only zero square matrices are both upper triangular and skew-symmetric. II. (10 points) Mark the following statements either "True" or "False". Circle T for true and F for False. No proof or explanation is needed. 1. (F) [3 , 5 , 2] and [6 , 10 , 5] are parallel. 2. (T) The converse and inverse of a statement are logically equivalent. 3. (F) The negation of " A and B " is "not A and not B ." 4. (F) Let A and B be two matrices. If AB = O , then A = O or B = O . 5. (F) Consider a system of m linear equations in n variables. If m < n , then the system always have in nitely many solutions. III.(15 points) Let x 1 , . . . , x m be a mutually orthogonal set of nonzero vectors in R n . Use induction to show that ± ± ± ± ± m i =1 x i ± ± ± ± ± 2 = m i =1 x i 2 Answer: Base step: When m = 1 , the statement is obviously true ( x 1 2 = x 1 2 ). 1

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Inductive step: Assume the statement is true when m = k , that is ± ± ± ± ± k i =1 x i ± ± ± ± ± 2 = k i =1 x i 2 .
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## This note was uploaded on 12/05/2011 for the course M 341 taught by Professor Hietmann during the Spring '08 term at University of Texas.

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exam1_sl - M341 Midterm Exam 1 Solution I.(15 points)...

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