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Tom O'Keeffe NAE HW6 6.1.15 a. yes, Ax <= b. b. Ax <= b, such that xi of (x1,x2,x3) where i =1,2,3 is changed one at time. Xi is increased until Ax is less than to equal to b. X1= 1200, increase of 200. X2= 650, increase of 150 X3= 450, increase of 100 X4= 500, increase of 100 c. Eliminating the column from the A matrix associated with the extinct species. Xi is increased until Ax is less than o equal to b. X2= increase of 650 X3= increase of 150 X4= increase of 150 d. Eliminating both column one and two, extenct species and then xi is increased until Ax is less than o equal to b. X3= increase of 150 X4= increase of 150 6.2.9.d A=((3.33 15920 10.333 | 7953)(-2.22 16.710 9.6210 | 0.965)(-1.5611 5.1792 1.6855 | 2.714)) S1=15920, S2=16.710, S3=5.1792 E1->E3, E3-> E1, E2-(2.222/-1.5611)E1->E2, E3-(3.333/-1.5611)E1->E3,E3-(1591.06/24.08184)E2->E3 A'x=((-1.5611 5.1792 -1.6855)(0 24.08184 7.21.2935)(0 0 -4764.0))=(2.714 4.827 4764.987) Backsubstitution yields x=(1 0.5 -1) 6.3.11 a. A^2=AA=((0 2 0)(0 0 3)(1/6 0 0)) A^3=AAA=((100)(010)(001)) Every three years the cycle repeats, period=3. b. 6000 initial beetle population of ages groups 1,2 and 3 over sequential years. Age group 1 6000->36000->12000->6000 Age group 2 6000->3000->18000->6000 Age group 3 6000->2000->1000->6000 c. A^-1=((0 2 0)(0 0 3)(1/6 0 0)) The aij matrix denotes the contribution of a single beetle to the next year population based on age. The inverse of aij = A^-1 denotes the number of beetles needed to produce a single beetle for the next years population based on the beetle's age. 6.5.3.a PA=LU, by making the row change E2->E3 and E3->E2 allows the permuted matrix (PA) to be solved by factoring. P=((1 0 0)(0 0 1)(0 1 0)) PA=((2 -1 1)(3 3 5)(3 3 9)) Ly=b, where L=((1 0 0)(1.5 1 0)(1.5 1 1)), y=(-1 1.5 4) Ux=y where U=((2 -1 1)(0 4.5 7.5)(0 0 -4)), X=(1 2 -1) 6.6.6.d A=((0.5 0.25 0 0)(0.35 0.8 0.4 0)(0 0.25 1 0.5)(0 0 1 -2)) L=((0.5 0 0 0)(0.35 0.625 0 0)(0 0.25 0.84 0)(0 0 1 -2.595)) U=((1 0.5 0 0)(0 1 0.64 0)(0 0 1 0.595)(0 0 0 1)) Ly=b, y=0.7 0.84 -0.845 0.541) Ux=y where x=(-0.0917 1.518 -1.16 0.541)