Question

MUID-TERM (40 marks) Instructions: Attempt ALL qtion Show COMPLETE working Question In geven electrical network, the equations for the events i... are given by 21-5+59 1 4+39-6-07 3,2,=-3.1 Use Gaussian cumination method to evaluate the recent (10 marks) Question 2 The mate of the system XXX given by (110 =10 11 1 0 1 Find the eigenvalues of A (& marks) Determine the corresponding cige vectors of 4 (12 marks) Question 3 inted Statesi imo e 1 DELL

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Answer 1

0.4. biven- 21, -i, + iz = 5.9 + fa- -i = 0.7 -3i + 2 12 = - 3.1 matrix- make an @gym argumented Let US 5.9 o et ! iz -3.1 - 3 R3 - R₂ + 3 R Rg - Ra elor 5.9 ia -2.25 O 13 5.75 3 (R3 R3 + 3 Rue) 5.9 i. 2. -2.25 ia iz 5 2.375 0 0

Hence, Restoring the original mataise - 5.9 -2.25 3 - 574 2.375 0 Hence, le 2 + z = 5.9 3 13 = -2.25 pa 11 slur 2.375 0,6 2.3 1}z=log 0_2. briven matrix - 9) A = Guven, AX= 2x → (A-TI) x = 0

3 O 1-1 O 1 0 -> I-n 0 1-3) R1 about -- Opening -> (1-9) ((1-0)2-0) -- (-1) + 0 = 0 => 11-093 +1.= 0

=> 1-03_ 32 11-9).+!=0 - 1-23- 37 +372 +1 = 0 73-372 +39-2=0 2=2 n=1 10 Hence, the eigen values of A are 7=2, 3 and est

b) Now, For eigen vectoas cones ponding to the eigen values calculated in part A- For 2= 2 - 1 x2 223 1-2 Xz=2 -xet x = 0 - Xq + 2z=0 xe, - &z=0 Hence, the conespondinding eigen vector, For n=2 1: X, = constant where K is a For n = 's I-5 22 dz

5 Xz -22 = -> x + x = 0 221 Ka = 22 +2₂=0 x = k x3=0 X, + Hence, the coresponding eigen vector, for n = 1/2 X3 K X2= -> X2 = K -ako -ak Hence since the the last eigen value is the eigen vector (X2 = X2) Hence, x = R · Yg= ées