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2006 Exam 2 Spring - PageZ of Problem 1(20 pts total Seiect...

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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+TX4-7xs =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 ‘flloafléiqx-esmz 0125*3 3 “‘3 i 1 "-1 ‘i Rf")?! 3 -3 I '1 “‘1 15 L9 0 _. M l-163 4.7 a 24,43ifi,x2+&,00—1+z-58 *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 ‘ ‘ . Miami-t FQJI 90‘3 X3 —2>_<4f Jr‘Q'X;-:— '9.) .— __ __ l .2 @ K4, oubi’i‘mv-H] X, r 9%. “gin ‘; 3 1': XI b?g-\r><q_ 6X4- I 5 {gt ‘1 1 5 0° .1, n25 L35»? x on x 4- + X N 1‘ 4X mNJ it wt 3‘ MOO-‘— (i \fi'fiw {$0 5%“ s—gw II I o w \£° 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)=(—I-A)C-I)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 —-—9-1 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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