This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 nonzero 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. ...
View
Full
Document
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.
 Spring '08
 Faculty

Click to edit the document details