MA110_2008FALL_EXAM2__[0] - MAIlOA FALL2008 EXAM #2 Name...

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Unformatted text preview: MAIlOA FALL2008 EXAM #2 Name Show all of your work and all of your answers in your bluebook. Please keep in mind that your explanations may not use chapter 2 material. When you are finished, hand in this question sheet with your bluebook. Please note that you should not discuss this exam with anyone before Saturday. 1) (12pts) Which of the following equations are linear? (E1) (:I:+y)2= (z—yr)2 (E2) 3z+4y—2z:7 (E3) 31 +_2$2+3x3+---+1{sz :0 (E4)$+3y—z=4z—y+z (E5)z2z—%y+(sin1)z:2 (E6)3:y—*yz=3:+y+z 2) (10pts) (a) Which (if any) of the following matrices are symmetric? (b) Which (if any) are upper triangular? (0) Which (if any) are lower triangular? (d) Which (if any) are diagonal? 010 147 100 000 A=[—10—1 B: 0—13 0': 000 D: 000 010 000 001 000 001 E2010 100 2m +4y+2az=—2 3) (15pts) Consider the system { —2:r — 3y — (a + 1)z:3 318+ (a.+ ?)'y+3az:—3 For what value(s) of a. (if any) is the system inconsistent? For what value(s) of a (if any) does the system have infinitely many solutions? 4) (ths) Suppose A, B and C are matrices whose sizes make the expression (B — C)A meaningful. Explain why it follows that BA — CA is also meaningful, then prove that BA—CA = (B— C)A. —2 1 15 5) (12pts) Determine whether or not the matrix A = [ 1 0 —6 ] can be written as a —1 1 12 product of elementary matrices. If it can, then do so (including a clear indication of the order of the product). If it can not, then clearly and concisely explain why not (citing chapter 1 theory as appropriate). 6) (24pts) Determine which of the following statements are true, and which are false. (Please read each statement carefully — some of them may be tricky. Be sure to treat general claims appropriately.) l 0 1 2 0 1 0 1 2 0 (a) 0 2 4 6 2 Z 0 1 2 3 1 0 0 1 3 1 0 0 1 3 1 0 0 2 5 3 0 0 0 —1 1 (b) A homogeneous linear system with more equations than unknowns has only the trivial solution. (c) If A and B are square matrices of the same size, then (A + B)2 : A2 + 2.43 + 32. (d) If A and B are square matrices of the same size, then tr(A + B) = tr(AT + B). a: 3x — y (e) There exists a 3x3 matrix A such that A y ] : [ 3y — z ] for all choices of :12, y and z. z 32 -— as f) If the first column of A has all zeros, then so does the first column of every product AB. g) If A and B are square matrices of the same size and AB = 0, then either A = U or B = 0. As always you should treat the “or” as an inclusive or.) h) It is possible for a linear system with 4 equations and 4 unknowns to have precisely 4 o ) If A is a non-zero symmetric matrix, then A is invertible. j) The product of symmetric matrices is symmetric. (k) Suppose A is the coefficient matrix for some linear system of equations. If the reduced 1 U 0 3 row echelon form of A is g (1] E] _11 , then the system has infinitely many solutions. 0 0 O 0 (1) If A is a 3x3 singular matrix, and matrix B may be obtained by performing 3 elementary row operations on A, then B is also singular. 1 7) (18pts) Suppose A is a 3x3 matrix such that the system Ari—2' : -—3 ] has infinitely many 2 solutions. Which of the following statements (if any) are certainly true, which (if any) are certainly false, and which (if any) have a truth value which cannot be determined from the given information? (Sl)A:E':[12 3] (S2)As=[12 3] (S3) A5: [1 2 3f has no solutions. 4 4 4 has a unique solution. has infinitely many solutions. (S4) A53 2 [ —2 6 ]T has a unique solution. (S5) A11? : [ —2 6 ]T has infinitely many solutions. (S6) A5 = [ —2 6 ]T has no solutions- (S?) Ax? = [ 0 U 0 ]T has a unique solution. (38) Azf = [ 0 0 0 ]T has infinitely many solutions. (SQ) A3? = [ U 0 0 ]T has no solutions. ...
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This note was uploaded on 02/27/2012 for the course CHEMISTRY/ CH/ECE/PH/ taught by Professor Faculty during the Spring '08 term at Cooper Union.

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MA110_2008FALL_EXAM2__[0] - MAIlOA FALL2008 EXAM #2 Name...

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