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Fall 2009 Midterm #1 Solutions

# Fall 2009 Midterm #1 Solutions - 0 9L 5C9)“ MATH1850/2050...

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Unformatted text preview: . 0 9L) 5C9)“ MATH1850/2050 809 Name: Page 1 of 6 (1) (6 marks) Solve the linear systems associated with the following matrices: 1 3 0 —1 4 1 3 —2 5 (2') 0 0 1 2 3 (12') 0 1 2 —2 0 0 0 0 0 0 0 1 —1 7"“ L} '35-‘ck X\ = 5 953’: 5 XL= O X3= 3-2.1} x3: "I x“: 1: watts ch GIL (2) (5 marks) Find conditions on k such that the following system has (i) exactly one solution, (ii) no solution, (iii) oo—niany solutions. 2:61 -- 2k\$2 4 3:61 -- (k2+2)\$2 = k+4 2. 1'. H Um. kw w [ 4 I4 7.. __——> mu“ 0 lot—Ska \L-‘L 1F 1L7; Sk-VL f O 22.3. I“: \Lf 11,1 Tag» 531 MM A SOLUTi9Q r—'—'—1 (HP '11: k:)_ (.21 0° -I’\Au~/ gnu-r Caps 1p k=\ (:21 NO 501.011.0105 MATH1850/2050 809 Name: Page 2 of 6 (3) (5 marks) Find the inverse of the following matrix (make sure you label all your row operations and state What A‘1 is explicitly): 1 0 0 1 0 0 2 0 A _ 2 1 1 1 —2 0 0 —1 U SE m MATHl850/2050 809 Name: Page 3 of 6 (4) (5 marks) Determine the value(s) of k for which A fails to be invertible 1 —3 —4 A: 2 —6 k—3 —2 k+1 5 A mus To m \Juemeuz {cc MA=O \ —3 -‘1 \ '3 -" MA= \ :L -c, k—S = o o k+§ [tau-22. -L \uu 5' o h—S '3: Lngzlu H I —.’> "f , 0 k4 _\$\: —QL"5XH§\ ‘0 L“ \u—‘iS. o 0 US 50 A FACLs To 5: Quart-ism 1W \4.= :L-g, (5) (4 marks) Solve for the matrix B in the following matrix identity ( show all steps, and mind the order of operations): (—1BT—312)—1 = [3 ‘4] 2 2 —2 c \ Rinses : —- TAK. N :3: 6" ”3:7. = ‘: l l 1 _\ 92 ADD 51:1 - _\_6\" _ [Z 7' '2- —\ ‘VL scALAL Honda»; ‘6“! 9.: 6‘- : [u ‘1. ”l a TAKE «LN-buses: l5 _ [(1 ﬂ] H a MATH1850/2050 809 Name: Page 4 of 6 (6) (3 marks) Find the second row, third column entry of the matrix prod- uct ABC Where —5 0 3 —2 3 0 2 0 CO H | .4; COHNDO (7) (3 marks) Use A‘1 = @adjﬂl) and det(A) = —20 to ﬁnd (A‘1)23, that is, the entry in row 2, column 3 of the inverse for the following matrix 2 1 3 —1 4 0 5 —2 1 —4 A = —2 2 —3 5 —3 0 1 0 1 0 0 2 0 0 3 My gee-emu = 1-1: (cm = inc“ 3+2 7. 3 -\ ‘1 2‘ C31=6\) o —z \—'~t =|1L MATHl850/2050 809 Name: Page 5 of 6 (8) (5 marks) True / False. Indicate whether the following statements are always True or sometimes False. (a) Every homogeneous linear system is consistent E/True False (b) If A, B are square matrices of same size, then (AB)2 = A2B2 True E/False (c) Every singular matrix has at least one row of zeros in its reduced row echelon form 3/ True False (d) If A is a square matrix with two proportional rows, then A is singular E/True False (e) If the linear system Ax = 0 has inﬁnitely many solutions, then the matrix A is invertible True E/False (f) A linear system of 4 equations in 3 unknowns must be inconsistent True 9/ False (g) The permutation (4,2,5,3, 1) is odd WNW?- True False (h) If A is an upper triangular matrix then AT is lower triangular True False (i) If det(AB) = 2, then A and B are invertible True False (j) The product of any two elementary matrices is an elementary matrix True E/False MATH1850/2050 809 Name: Page 6 of 6 (9) (4 marks) (a) Show that if A and B are symmetric matrices of same size, then so is A — kB. lLHouJ AEA, e921; , mm (A-kﬁf; Aug, (A-¥P>Y= AT—QLB)T= A— Li; a A—MS so A—kﬁ is synmnlc. (b) Suppose that A and B are invertible square matrices of the same size. Prove that (ABA—1)_1 = AB‘1A_1. (AeK‘X Aw“) = AuA-‘mm' = A51 Q‘PI‘ = ALMJ‘M' = ALA“ -: AA" . «.3: 2 smog ABA" ts sauna? (AI-5A" "= As‘pi‘ . ...
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Fall 2009 Midterm #1 Solutions - 0 9L 5C9)“ MATH1850/2050...

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