MAT 205 Midterm, Spring 2012

MAT 205 ivi al argument or a counterexample. Do not guess; the number of Wrong answers will be subtracted from the number of right answers, with the possibility of a negative score! 1) IfA and B are invertible matrices and X4 = B, then X= 4"'B (. “E) 2) IfA is a (3x3) matrix and det (A) #0, then det [adj (4)]=(det 4)? ( wend...) 3) For the system given below, a necessary and sufficient condition for the existence of a solution is: 3a=c—b (vs 1. x+3y+5z=¢ k-2 3 4) The matrix " " | is invertible for any real number “x? (on...). 1 0 5) The matrix ]0 0 0 1 or Oo >» a 3 | is in reduced-row echelon form (uEt..) 4 6) Ifthe matrix B is obtained by multiplying a column ofa matrix A by 9, then det(B) = 9det(A) (.20....), x+y-z=1 7) The system }2x+3y+kz=3 has infinitely many solutions for “k=2” (oh... x+ky+3z=2 8) If the matrix A is invertible, then det(A "A)> 0(.. fy, 9) Suppose 4 and B are invertible matrices and X is a matrix such that AXB = 4 + B, then X=A'+ B07... 10) A diagonal matrix with a “0” on its main diagonal is not invertible G XT... XA=8 XAR = BAY = = BAT ci (a) P=yaay te » > A O)- HW Ks aE BWP, baakto 4% ~~ At 4 ad won ») = Se) 4 2 a\firr a\\. Bil iy 43 5 © : 2 orn: 6 (exshaw of > Aichi eo a _ tebe = od 0 ct) ~ Gaze —— t vy EO - dolexnu nan ; h K2 2 i abe 4 ub p. 1 of 6 MAT 205 II. (10 Points) Suppose uy ‘ Uy and uz are three nonzero vectors such that uy, Lu, ; uy luz and Uy Luz (i.e they are orthogonal to each other), and suppose that Sow a on ; ane “ir b) If u =(1;—2;3), uy = (1;2;1)and U3 = (—8;2:4), are orthogonal to each other and by using the result of part W=(13;-4;7) = au; + bu, tems ~ ~ 7 : = 2) Wet = AR sbs+ Cus, | =p ac wo al uw “4 Se Ko" b eel ¢ ») . u,= O U,. 4 =O U,.u,=0 show that these three vectors (a) find a, b and c such that a T4Q . 20 (13; -w)- (3 9)_ HT Sd atate Til. (20 Points) Solve the following system for “y” xX+yt+z+w=2 X+2y4+3z+4w=2 2x+3y+5z+9w=2 xX+y+2z+7w=2 A A 4 | _s 5s 4 9 4 4 % 4H) os 4 9 Bes MAT 205 using Cramer’s rule: 244 S24 ou 3 A\, 3 + 218 o1 4 p. 3 of 6 ole — (FLAPS Ts,3;0)o (252, ve) dros] | (3,15; 8) : ©) |u—v+ w| | oe ($93) - a Rl saps sil De . _ 5a 4] oe dh Wear Fo Tomo eat = (9xd)- yaaa | _ fy (ara) (sy 5) || al] (x8) ai fs 3 ON ey \ ; Ped ») \\ or -¥ | : (ns (5:238) os) 34. ub , oe ~ i Te Ty? TH A 4-4 (a) a u oe 2 |e c) R.ad\WN= | 2 - A e ) j 5 tht? \ A ) yomh Wel@ 34 pa \ 1 2 3 V. (20 Points) Consider the Square matrix: 4 = 2 3 ’ Le 7 a) Find the determinant of 4: |A| b) Find the adjoint of A: adj(4) ¢) Verify that 4.(adjA) =|A|I where Zis the identity matrix, d) Use adj(4) to find 47! MAT 205 | -lOo 4+ >|, 4 -% 42-1 p. 5 of 6 2) Consider a skew symmetric (nxn) matrix 4 (ie. 47 =~A), where n is odd. Nae that det (4) =0, b)F or an invertible (nx xn) matrix A, show that A. adj(A) = det(A). J, where I is the (xn) identity matrix. (-1y'dek (a) = bth = (ro)_f a ~ ur = det A aoldt Re p. 6 of 6