Question

Question 1 3 2 2 (a) Find the LU decomposition for the matrix610 15 -3 Then use L U to solve the system AX-13-2 10 for X-xi, xx Question 2 A mass a is suspended by three cables attached at the thrce points B. C, and D as shown in the figure Let T:·T> and T1, be the tensons in tbe three cables AB, AC, and AD, respectively If the mass m is stationary, the sum of the tension components in the x, in the y, and in the z directions must each be zero This requirement gives the following three equations 37, 4r, Find Ti. T, and Ti, using Gauss Seidel and the initial values of 140, 40, 401, for a mass of I kg considering a gravity acceleration of 9.8 m/s Tip: don't for get to verify if it is in dominant diagonal form.

Answer 1

Ne have, 3 1O 5-3 1 CWhere 1 O O amd 21 L31 L i 3 2 rUI 13 22 U, 23 0 C mulHiplyin out LU amd Setting the answer equal to A gives

12 3 2 21 12 6 1 0 3 31 u e begin b ing along the top To to See thab 11 Z3 12 -2 13-2 Now Consider the Second γου 21 12 U22 - Il

2 3 Coming t the Last row, 31 VI 15 32 22 32 (7) - -3

(s)(2)) U33- 1 he have shown that 2 LU S -3 1 2 o 0 구 O 0-3 and this İSan LU decomposition of A Now where 4 That gves, 1 0O 2 -21 O 2 уз

hich can be Solved by fofuard Subs titution from the top equation, we See thae The middle equation states that 2 2. Finally the botto line says that ow wc have found V We Anish the pocedure bj solvin for X. That is We Solve, 3 3 2 2 2 23 by using back substitt StarHinthe bo ttom, eution w See that

he iddle equation implies hat +12-6(3):4 top equation give S 3x,-2X2 +2x3 = 3

x1- 2 -) 3지-44+6=3 37i 3x, 3- 23 2 1-23

23 L 91