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Question

Show that each one of the following systems of linear equation is inconsistent:
(i) 2x + 5y = 7
6x + 15y = 13

(ii) 2x + 3y = 5
6x + 9y = 10

(iii) 4x − 2y = 3
6x − 3y = 5

(iv) 4x − 5y − 2z = 2
5x − 4y + 2z = −2
2x + 2y + 8z = −1

(v) 3x − y − 2z = 2
2y − z = −1
3x − 5y = 3

(vi) x + y − 2z = 5
x − 2y + z = −2
−2x + y + z = 4

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Solution


(i) The given system of equations can be expressed as follows:AX=B Here, A=25615, X=xy and B=713Now, A =25615 =30-30 =0Let Cij be the cofactors of the elements aij in A =aij. Then,C11=-11+1 15=15,C12=-11+2 6 =-6C21=-12+1 5=-5,C22=-12+2 2=2adj A=15-6-52T =15-5-62adj A B=15-5-62713 =105-65-42+26 =40-16 0Hence, the given system of equations is inconsistent.


(ii) The given system of equations can be expressed as follows:AX=B Here, A=2369, X=xy and B=510Now, A =2369 =18-18 =0Let Cij be the cofactors of the elements aij in A =aij. Then,C11=-11+1 9=9 ,C12=-11+2 6 =-6 C21=-12+1 3=-3, C22=-12+2 2=2adj A=9-6-32T =9-3-62adj A B=9-3-62510 =45-30-30+20 =15-10 0Hence, the given system of equations is inconsistent.


(iii) The given system of equations can be expressed as follows:AX=BHere, A=4-26-3, X=xy and B=35 A =4-26-3 =-12+12 =0Let Cij be the cofactors of the elements aij in A =aij. Then,C11=-11+1 -3=-3, C12=-11+2 6 =-6C21=-12+1 -2=2, C22=-12+2 4=4adj A=-3-624T =-32-64adj A B=-32-6435 =-9+10-18+20 =12 0Hence, the given system of equations is inconsistent.


(iv) The given system of equations can be written as follows:AX=B Here, A=4-5-25-42228, X=xyz and B=2-2-1 A =4-5-25-42228 =4-32-4+540-4-2(10+8) =-144+180-36 =0Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+1-4228 =28, C12=-11+25228 =-36, C13=-11+35-422=18C21=-12+1-5-228 =36 , C22=-12+2 4-228 =36 , C23=-12+34-522=-18C31=-13+1-5-2-42 =-18, C32=-13+24-252 =-18, C33=-13+34-55-4=9adj A=28-36183636-18-18-189T = 2836-18-3636-1818-189adj AB= 2836-18-3636-1818-1892-2-1 =56-72+18-72-72+1836+36-9 =2-12663 0Hence, the given system of equations is consistent.


(v) The given system of equations can be written as follows:AX=B Here, A=3-1-202-13-50, X=xyz and B=2-13 A =3-1-202-13-50 =30-5+10+3-2(0-6) =-15+3+12 =0Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+12-1-50 =-5, C12=-11+20-130 =-3, C13=-11+3023-5=-6C21=-12+1-1-2-50 =10, C22=-12+2 3-230 =6, C23=-12+33-13-5=12C31=-13+1-1-22-1 =5, C32=-13+23-20-1 =3, C33=-13+33-102=6adj A=-5-3-610612536T = -5105-363-6126adj AB=-5105-363-61262-13=-10-10+15-6-6+9-12-12+18=-5-3-6 0Hence, the given system of equations is consistent.


(vi) The given system of equations can be written as follows:AX=B Here, A=11-21-21-211, X=xyz and B=5-24 A =11-21-21-211 =1-2-1-11+2-2(1-4) =-3-3+6 =0Let Cij be the cofactors of the elements aij in Aaij. Then,C11=-11+1-2111 =-3 , C12=-11+211-21 =-3, C13=-11+31-2-21=-3C21=-12+11-211 =-3, C22=-12+2 1-2-21 =-3 , C23=-12+311-21=-3C31=-13+11-2-21 =-3, C32=-13+21-211 =-3, C33=-13+3111-2=-3adj A=-3-3-3-3-3-3-3-3-3T =-3-3-3-3-3-3-3-3-3adj AB=-3-3-3-3-3-3-3-3-35-24=-15+6-12-15+6-12-15+6-12=-21-21-21 0Hence, the given system of equations is consistent.

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