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Question

Examine the consistency of the system of equations3xy2z=2,2yz=1,3x5y=3

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Solution

Given system of equations
3xy2z=2
2yz=1
3x5y=3
This can be written as
AX=B
where A=312021350,X=xyz,B=213

Here, |A|=3(05)+1(0+3)2(06)
|A|=0
Since, |A|=0
Hence, the system of equations has either infinitely many solutions (consistent) or no solution (inconsistent).

We need to find (adjA)B
C11=(1)1+12150
C11=05=5

C12=(1)1+20130
C12=(0+3)=3

C13=(1)1+30235
C13=06=6

C21=(1)2+11250
C21=(010)=10

C22=(1)2+23230
C22=0+6=6

C23=(1)2+33135
C23=(15+3)=12

C31=(1)3+11221
C31=1+4=5

C32=(1)3+23201
C32=(30)=3

C33=(1)3+33102
C33=60=6

Hence, the co-factor matrix is C=53610612536

adjA=CT=51053636126

Now, (adjA)B=51053636126213

=1010+1566+91212+18

(adjA)B=536
Since, (adjA)BO
Hence, the system of equations is inconsistent.

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