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Unformatted text preview: PageZ of? Problem 1(20 pts total) Seiect the best answer to each of the following and circle your choice. A. f A,B,and C are all nxn matrices, which of these is always true?
AB+CA = A(B+C) (A+B)TC = are + BTC (AB)T = ATBT 4) (AB)2 = A2132 5) all of the above are aiways true / B. If D is an nxn diagonal matrix, and B is a matrix, then the weakest condition we can put on B so that
: D = DB is 3) B is an rpm scalar matrix
4) B =11. The equation 2x+3y—z = 5 describes a
1 plane perpendicular to the vector (2,3 ,I)
2) line through the point (2,0,4)
3) plane through the point (2,3,—1)
ML 4 _4 plane perpendicuiar to the vector (0,0,5)
‘ line perpendicular to the vector (2,3,1) D. The system of linear equations AX = B might be inconsistent if
1) B = [0] J” rank of A = number of rows in A
rank of A 2 number of columns in A row reduced form of A does not have any zero rows ._ l ___\ , I
( as) ., s A
E. If both A and B are invertible matrices and AB lA'lX‘= B'1 then the simplest expression we can write for X
is A (49(97wa : fr} ,
/r’ l X = I
or x = AB(BA)'1 x. ; ﬁr
X = (B‘IA 2 J.
4) X = B‘IA‘ F. If the columns of A are linearly dependent, then
1) rank of A will be zero
if A is square it will be nonsingular
l) the system of linear equations AX=B cannot have a unique solution
the rows of A will be linearly dependent f G. If A is an nxn matrix and U is the row reduced echelon form of A then
1) det(A) = det(U)
2) eiqenvalues of A and U are the same
A‘ = U
/’ none of the above is always true Page 3 of 7 3,2,
(1)5 3249
— 5674
3 det
)det(A) —1285
13x
6 3149 2 3249
— 8231 — 5674
2) det(A)det—1 3 3 5 )det(A)det—1 2 8 5
463x #2 413x grime of these I. If, for a particular numerical value of on, the matrix A—ocI is reduced by row operations to the matrix
/ 1 0 0
0 1 0 , we could conclude that
0 0 1 l 0 0
l) the eigenveetors corresponding to or are [0],[1], and[0]
0 0 l 0
2) the eigenvector corresponding to a is [0]
0 f ' 0L is not an eigenvalue forA
'H a O is an eigenvalue for A OOH 0201:
1304:
0015: HNL»
\__._.___./ J. If the reduced augmented matrix for the system of linear equations AX=B is [ I) the system will have one unique solution
2) the system will have an inﬁnite number of solutions with five arbitrary parameters
I. 3 the system will have an inﬁnite number of solutions with four arbitrary parameters
’“ A the system will have an inﬁnite number of solutions with three arbitrary parameters
the system will have an inﬁnite number of solutions with two arbitrary parameters Page 4 of? Problem 2: (15pts total) Use Gaussian elimination on the augmented matrix to solve the system of
equations below. If the system is inconsistent, indicate clearly why. If there is a unique solution,
express it as a column vector. If there are an inﬁnite number of solutions, write the general form for
solutions as a linear combination of column vectors. 3x1—3x2+X3+TX47xs =13 ”“3 "Mt ' 23,
Xl—Xz +3X4—4X5=7 ? I]: “If
2x]+2x2—X3—4X4+3Xs=5 rq Jr"? 1,1 0 3 4 7
r ~13 ‘ﬂloaﬂéiqxesmz 0125*3
3 “‘3 i 1 "1 ‘i Rf")?! 3 3 I '1 “‘1 15 L9 0
_. M
l163 4.7 a 24,43iﬁ,x2+&,00—1+z58
*1 2 1 4t 3 'L 1
fr :1 :2.) "Ag is" '4 WM?! \ .1 l I i '"l
o o 1—2 5 ~92 ﬂ 0 O ‘ "2 5 FE)
7 (9 O O O C7
O o o o o o
Fat—Rb
‘ ‘ . Miamit FQJI 90‘3 X3 —2>_<4f Jr‘Q'X;:— '9.) .— __ __ l .2
@ K4, oubi’i‘mvH] X, r 9%. “gin ‘; 3
1': XI b?g\r><q_ 6X4 I 5 {gt ‘1
1 5
0° .1,
n25
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on
x 4
+ X
N 1‘
4X mNJ
it wt
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\£° e Pagcﬁuf'! 3 —1 2 2 7
a) Solvethema' .Iuaﬁon A+BX+CX=D forthematrixX. :: I/
I q» ’9 E 5]
a J, 2‘? x2 2 I 2 1 2 1 1 I 4 2 5
Problem 3: (20pm total) Let A: 2 , B: , C= , and D: . 7, \ m+9c i ‘ 2C*‘"qc’) ”c
. .._P l .42 ngc’rbc a+¢1¢¢rrl ’15.; :9:
2b we) 2 ’2 c 4?...
I3
b 1010‘
‘n mad! — r") I 5 I ‘
A B A'1 x
b) The Inverse of the 4 x 4 partitioned matrix is given by C4 , where X 15 a 2x 2 mamx. Find x. (g '3 E} % A 36“}, [06) [£6 :1 g  kc,
@ K '; (5 b) Solve the system of Equations AX : B for the variable x1 . o 4 O o
’X?;de/PML 2 _l 2 doth M,lo l ‘ O Pageéof? {C
22 Page'lof 7
0 _] I (A rﬁl): —7\ .‘ _._ '. — l
Problem 5: (25pts total)Let A = 0 —1 0 . 0 Jr‘ ‘7\. 0 f
—1 1 o _ ‘ I __ a 21) Find the eigenvalues and eigenvectors of A det(A~m1)=(—IA)CI)dz£ }\ = I —i —l — I o
O “2 0 O Si:
5’ ’ >
——1 1 ,3 0 r L”'
£2???) 1 I l o 92/ l 1 I o
“—7 0 *1 o O ——91 g l D D 7‘?
U 0 o O a C) 0
9} ; "'1 I “"l I ’3 1 ' i
0 O 0" "_—'1) "1 I
X0" — o o
[(K —)<21—)<5 7:5 O x‘ bx) 1)) Find In(A+ 21). (A .1» DI) ﬁbra) (2.7911. 5][
6: (1 7Q 4+7‘F4ﬁ— —‘}5( 7\) 9:2 thaws): )c, j I ,9 n
1 IO ‘00
‘0‘)ij o tﬁbom ‘
0 0 1 0 0‘ C) ‘0
o lo ...
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 Spring '08
 BARR

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