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Question

Solve the following system of equations by matrix method:
(i) x + yz = 3
2x + 3y + z = 10
3xy − 7z = 1

(ii) x + y + z = 3
2xy + z = − 1
2x + y − 3z = − 9

(iii) 6x − 12y + 25z = 4
4x + 15y − 20z = 3
2x + 18y + 15z = 10

(iv) 3x + 4y + 7z = 14
2xy + 3z = 4
x + 2y − 3z = 0

(v)
2x-3y+3z=101x+1y+1z=103x-1y+2z=13

(vi) 5x + 3y + z = 16
2x + y + 3z = 19
x + 2y + 4z = 25

(vii) 3x + 4y + 2z = 8
2y − 3z = 3
x − 2y + 6z = −2

(viii) 2x + y + z = 2
x + 3yz = 5
3x + y − 2z = 6

(ix) 2x + 6y = 2
3xz = −8
2xy + z = −3

(x) xy + z = 2
2xy = 0
2yz = 1

(xi) 8x + 4y + 3z = 18
2x + y +z = 5
x + 2y + z = 5

(xii) x + y + z = 6
x + 2z = 7
3x + y + z = 12

(xiii) 2x+3y+10z=4, 4x-6y+5z=1, 6x+9y-20z=2;x, y, z0

(xiv) xy + 2z = 7
3x + 4y − 5z = −5
2x y + 3z = 12

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Solution

(i)
Here,A=11-12313-1-7 A =11-12313-1-7 =1-21+1-1-14-3-1(-2-9) =-20+17+11 =8Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+131-1-7 =-20 , C12=-11+2213-7=17, C13=-11+3233-1=-11C21=-12+11-1-1-7 =8 , C22=-12+2 1-13-7 =-4, C23=-12+3113-1=4C31=-13+11-131 =4 , C32=-13+21-121 =-3, C33=-13+31123=1adj A=-2017-118-444-31T = -208417-4-3-1141A-1=1Aadj A=18-208417-4-3-1141X=A-1Bxyz=18-208417-4-3-11413101xyz=18-60+80+451-40-3-33+40+1xyz=182488x=248, y=88 and z=88 x=3, y=1 and z=1

(ii) Here,A=1112-1121-3 A =1112-1121-3 =13-1-1-6-2+1(2+2) =2+8+4 =14Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+1-111-3 =2 , C12=-11+2212-3 =8, C13=-11+32-121=4C21=-12+1111-3 =4, C22=-12+2 112-3 =-5, C23=-12+31121=1C31=-13+111-11 =2 , C32=-13+21121 =1, C33=-13+3112-1=-3adj A=2844-5121-3T = 2428-5141-3A-1=1Aadj A=1142428-5141-3X=A-1Bxyz=1142428-5141-33-1-9xyz=1146-4-1824+5-912-1+27xyz=114-162038x=-1614, y=2014and z=3814 x=-87, y= 107and z=197

(iii) Here,A=6-1225415-2021815 A =6-1225415-2021815 =6225+360+1260+40+25(72-30) =3510+1200+1050 =5760Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+115-201815 =585, C12=-11+24-20215 =-100 , C13=-11+3415218=42C21=-12+1-12251815 = 630, C22=-12+2 625215 =40, C23=-12+36-12218=-132C31=-13+1-122515-20 =-135, C32=-13+26254-20 =220, C33=-13+36-12415=138adj A=585-1004263040-132-135220138T = 585630-135-1004022042-132138A-1=1Aadj A =15760585630-135-1004022042-132138X=A-1Bxyz=15760585630-135-1004022042-1321384310xyz=157602340+1890-1350-400+120+2200168-396+1380xyz=15760288019201152x=28805760, y=19205760and z=11525760 x= 12, y= 13and z= 15

(iv) Here,A=3472-1321-3 A =3472-1321-3 =33-3-4-6-6+7(2+2) =0+48+28 =76Let Cij be the cofactors of elements aij in Aaij. Then,C11=-11+1-131-3 = 0, C12=-11+2232-3 =12, C13=-11+32-121=4C21=-12+1471-3 = 19 , C22=-12+2 372-3 =-23 , C23=-12+33421=5C31=-13+147-13 =19, C32=-13+23723 =5, C33=-13+3342-1=-11adj A=012419-235195-11T = 0191912-23545-11A-1=1Aadj A =1760191912-23545-11X=A-1Bxyz=1760191912-23545-111440xyz=1760+76+0168-92+056+20+0xyz=176767676x=7676, y=7676 and z=7676 x= 1, y= 1 and z=1


(v)
Let 1xbe a, 1ybe b and 1zbe c.Here,A=2-331113-12 A =2-331113-12 =22+1+32-3+3(-1-3) =6-3-12 =-9Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+111-12 =3, C12=-11+21132 =1 , C13=-11+3113-1=-4C21=-12+1-33-12 =3 , C22=-12+2 2332 =-5, C23=-12+32-33-1=-7C31=-13+1-3311 =-6 , C32=-13+22311 =1, C33=-13+32-311=5adj A=31-43-5-7-615T = 33-61-51-4-75A-1=1Aadj A =1-933-61-51-4-75X=A-1Babc=1-933-61-51-4-75101013abc=1-930+30-7810-50+13-40-70+65abc=1-9-18-27-45 x=1a=-9-18, y=1b=-9-27and z=1c=-9-45 x=1a= 12, y=1b= 13and z=1c= 15

(vi)

Here,A=531213124 A =531213124=54-6-38-3+1(4-1)=-10-15+3=-22Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+11324 = -2, C12=-11+22314 =-5 , C13=-11+32112=3C21=-12+13124 =-10 , C22=-12+2 5114 =19, C23=-12+35312=-7C31=-13+13113 =8 , C32=-13+25123 =-13 , C33=-13+35321=-1adj A=-2-53-1019-78-13-1T = -2-108-519-133-7-1A-1=1Aadj A =1-22-2-108-519-133-7-1X=A-1Bxyz=1-22-2-108-519-133-7-1161925xyz=1-22-32-190+200-80+361-32548-133-25xyz=1-22-22-44-110x=-22-22, y=-44-22and z=-110-22 x= 1, y= 2 and z= 5

(vii)
Here,A=34202-31-26 A =34202-31-26 =312-6-40+3+2(0-2) =18-12-4 =2Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+12-3-26 = 6, C12=-11+20-316 =-3 , C13=-11+3021-2=-2C21=-12+142-26 = -28 , C22=-12+2 3216 =16 , C23=-12+3341-2 =10C31=-13+1422-3 =-16, C32=-13+2320-3 =9, C33=-13+33402=6adj A=6-3-2-281610-1696T = 6-28-16-3169-2106A-1=1Aadj A=126-28-16-3169-2106X=A-1Bxyz=126-28-16-3169-210683-2xyz=1248-84+32-24+48-18-16+30-12xyz=12-462x=-42, y=62and z=22 x=-2, y= 3 and z=1


(viiii)Here,A=21113-131-2 A =21113-131-2 =2-6+1-1-2+3+1(1-9) =-10-1-8 =-19Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+13-11-2 =-5, C12=-11+21-13-2 =-1, C13=-11+31331=-8C21=-12+1111-2 = 3, C22=-12+2 2-13-2 =-7, C23=-12+32131=1C31=-13+1113-1 =-4, C32=-13+2211-1 =3, C33=-13+32113=5adj A=-5-1-83-71-435T = -53-4-1-73-815A-1=1Aadj A=1-19-53-4-1-73-815X=A-1Bxyz=1-19-53-4-1-73-815256xyz=1-19-10+15-24-2-35+18-16+5+30xyz=1-19-191919x=-19-19, y=19-19 and z=19-19 x= 1, y= 3 and z=-1


(ix) Here,A=26030-12-11 A =26030-12-11 =20-1-63+2+0(-3+0) =-2-30 =-32Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+10-1-11 =-1 , C12=-11+23-121 =-5 , C13=-11+3302-1=-3C21=-12+160-11 = -6 , C22=-12+2 2021 =2 , C23=-12+3262-1=14C31=-13+1600-1 =-6 , C32=-13+2203-1 =2 , C33=-13+32630=-18adj A=-1-5-3-6214-62-18T = -1-6-6-522-314-18A-1=1Aadj A =1-32-1-6-6-522-314-18X=A-1Bxyz=1-32-1-6-6-522-314-182-8-3xyz=1-32-2+48+18-10-16-6-6-112+54xyz=1-3264-32-64x=64-32, y=-32-32 and z=-64-32 x=-2, y= 1 and z= 2

(x) Here,A=1-112-1002-1 A =1-112-1002-1 =11-0+1-2-0+1(4-0) =1-2+4 =3Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+1-102-1 =1, C12=-11+2200-1 =2, C13=-11+32-102=4C21=-12+1-112-1 = 1, C22=-12+2 110-1 =-1, C23=-12+31-102=-2C31=-13+1-11-10 =1, C32=-13+21120 =2, C33=-13+31-12-1=1adj A=1241-1-2121T = 1112-124-21A-1=1Aadj A =111112-124-21X=A-1Bxyz=131112-124-21201xyz=132+14+28+1xyz=11369x=33, y=63 and z=93 x= 1, y= 2 and z= 3

(xi) Here,A=843211121 A =843211121 =81-2-42-1+3(4-1) =-8-4+9 =-3Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+11121 = -1, C12=-11+22111 =-1, C13=-11+32112=3C21=-12+14321 =2, C22=-12+2 8311 =5, C23=-12+38412=-12C31=-13+14311 =1, C32=-13+28321 =-2 , C33=-13+38421=0adj A=-1-1325-121-20T = -121-15-23-120A-1=1Aadj A =1-3-121-15-23-120X=A-1Bxyz=1-3-121-15-23-1201855xyz=1-3-18+10+5-18+25-1054-60xyz=1-3-3-3-6x=-3-3, y=-3-3and z=-6-3 x= 1, y= 1 and z= 2


(xii)Here,A=111102311 A =111102311 =10-2-11-6+1(1-0) =-2+5+1 =4Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+10211 = -2, C12=-11+21231 =5, C13=-11+31031 =1C21=-12+11111 =0, C22=-12+2 1131 =-2, C23=-12+31131=2C31=-13+11102 =2, C32=-13+21112 =-1, C33=-13+31110=-1adj A=-2510-222-1-1T = -2025-2-112-1A-1=1Aadj A =14-2025-2-112-1X=A-1Bxyz=14-2025-2-112-16712xyz=14-12+0+2430-14-126-14-12xyz=14124-20x=124, y=44and z=-204 x= 3, y= 1 and z= -5


(xiii) Let 1xbe a, 1ybe b and 1zbe c.Here,A=23104-6569-20 A =23104-6569-20 =2120-45-3-80-30+10(36+36) =150+330+720 =1200Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+1-659-20 =75, C12=-11+2456-20 =110, C13=-11+34-669=72C21=-12+13109-20 =150, C22=-12+2 2106-20 = -100, C23=-12+32369=0C31=-13+1310-65 =75, C32=-13+221045 =30, C33=-13+3234-6=-24adj A=7511072150-10007530-24T = 7515075110-10030720-24A-1=1Aadj A =11200 7515075110-10030720-24X=A-1Babc=11200 7515075110-10030720-24412abc=11200300+150+150440-100+60288-48abc=11200600400240x=1a=1200600, y=1b=1200400 and z=1c=1200240 x= 2, y=3 and z= 5


(xiv) Here,A=1-1 23 4-52-1 3 A =1-1 23 4-52-1 3 =112-5+19+10+2(-3-8) =7+19-22 =4Let Cij be the cofactors of elements aij in A=aij. Then,C11=-11+14-5-13=7, C12=-11+23-523=-19, C13=-11+3342-1=-11C21=-12+1-12-13=1 , C22=-12+2 1223=-1 , C23=-12+31-12-1=-1C31=-13+1-124-5=-3, C32=-13+2123-5=11, C33=-13+31-134=7adj A=7-19-111-1-1-3117T =71-3-19-111-11-17A-1=1Aadj A =1471-3-19-111-11-17X=A-1Bxyz=1471-3-19-111-11-177-512xyz=1449-5-36-133+5+132-77+5+84xyz=148412x=84, y=44 and z=124 x= 2, y= 1 and z=3.

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