MA110_2006FALL_EXAM2 - (d For what value(s of a(if any does the:system below have no solutions Make sure that you fully justify aU of your answers

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MAllO FALL2006 EXAM2 SHOW YOUR WORK! Your explanations may cite (without proof) any results from Anton, Chapter 1. A"QY additional theorems of linear algebra that yon may happen to know may not be used without proof. 2x 1 + 3x 2 - X3=0 .' 5xI + X2 + 2X3=0 -Xl + 5X2 - 4X3=0 (b) (a) I} (lOpts) Which of the following are linear equations in Xli $2 and X3? (a)x,+x~+:r,=4 (b)x,-7x3+1r=0 (C)3X,-X2+ 5X 3=-2 ( d) v'3x1 - v2x 2 + e 2 x3 ~ 0 (e) .L + 1.. - .L = I ~l %2 2;3 2) (12pts) Solve tbe following systems: 8x, + 21x2 + 3X3=7 -3xI + 9X2=-3 6x, + 9X2 + 3X3=7 3) (lOpts) True or false. If the first column of matrix A has all zeros, then so does the first column of every product AB. Make sure that you justify your answer. (This is a Discussion Discovery question frqm Anton.) 4) (16pts) (a) For what valuers) of a (if any) does the system below have a unique solution? (b) For what value(s) of a (ifany) does the system have more than one, but finiteLy many solutions? (c) For what valuers) of
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Unformatted text preview: (d) For what value(s) of a (if any) does the:system below have no solutions? Make sure that you fully justify aU of your answers. o-3x + 2az=-.a 2x + (a'- 'I)y + 3az=6a- 2-'X + 3az=2a 5) (lOpts) True or false. If A2 + 1 4 = 04X4, then A must be an invertible 4 x 4 matrix. Make sure that you justify your answer. 6) (l4pts) Determine whether or not [~ ~ ~] is invertible. if it is, find it.s inverse. • 1 0 2 7) (16pts) Wr,.te [~I O~-~I] as a product of elementary matrices. [ ' 0 0 0] [1 0] T [1 0 0-;"]-I [1 0 0 0] T 8100 0100 0100 0001 8) (12pts) Let A, = 0 0 1 0 1 0 0 0 1 0 0 1 0 • 00010-3010001 0100 [ ~ ~ ~ ~]-, [~ ~ ~ ~] [~ ~ ~ ~] [~ ~ ~ ~] T 5 7 A, = 0 0 I 0 0 I 0 0 ,2 0 0 1 0 ' and A = Al A,. 0001 100000010001 Determine all 4 x 1 maLrices b = [ ~ ] such that Ax = b has a Imique solution. Make sure that you fully justify your answer. ,...
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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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