3
Question
Examine the consistency of the system of equations
3x−y−2z=2,2y−z=−1,3x−5y=3
Examine the consistency of the system of equations
3x−y−2z=2,2y−z=−1,3x−5y=3
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Solution
The given system is 3x−y−2z=20x+2y−z=−1 and 3x−5y+0z=3
Which can be written as AX=B where
A=⎡⎢⎣3−1−202−13−50⎤⎥⎦,X=⎡⎢⎣xyz⎤⎥⎦ and B=⎡⎢⎣2−13⎤⎥⎦
Here, |A|=∣∣
∣∣3−1−202−13−50∣∣
∣∣=3(0−5)+1(0+3)−2(0−6)=−15+3+12=0
∴ A is a singular matrix
Therefore, nothing can be said about consistency as yet. So, we compute (adj A)B.
Cofactors of A are
A11−5,A12=−3,A13=−6,A21=10,A22=6,A23=12A31=5,A32=3,A33=6
adj(A)=⎡⎢⎣−5−3−610612536⎤⎥⎦T=⎡⎢⎣−5105−363−6126⎤⎥⎦
(adj A)B=⎡⎢⎣−5105−363−6126⎤⎥⎦⎡⎢⎣2−13⎤⎥⎦=⎡⎢⎣−10−10+15−6−6+9−12−12−18⎤⎥⎦=⎡⎢⎣−5−3−6⎤⎥⎦≠0
Thus, the solution of the given system of equations does not exist,
Hence, the system of equations is inconsistent.
The given system is 3x−y−2z=20x+2y−z=−1 and 3x−5y+0z=3
Which can be written as AX=B where
A=⎡⎢⎣3−1−202−13−50⎤⎥⎦,X=⎡⎢⎣xyz⎤⎥⎦ and B=⎡⎢⎣2−13⎤⎥⎦
Here, |A|=∣∣
∣∣3−1−202−13−50∣∣
∣∣=3(0−5)+1(0+3)−2(0−6)=−15+3+12=0
∴ A is a singular matrix
Therefore, nothing can be said about consistency as yet. So, we compute (adj A)B.
Cofactors of A are
A11−5,A12=−3,A13=−6,A21=10,A22=6,A23=12A31=5,A32=3,A33=6
adj(A)=⎡⎢⎣−5−3−610612536⎤⎥⎦T=⎡⎢⎣−5105−363−6126⎤⎥⎦
(adj A)B=⎡⎢⎣−5105−363−6126⎤⎥⎦⎡⎢⎣2−13⎤⎥⎦=⎡⎢⎣−10−10+15−6−6+9−12−12−18⎤⎥⎦=⎡⎢⎣−5−3−6⎤⎥⎦≠0
Thus, the solution of the given system of equations does not exist,
Hence, the system of equations is inconsistent.
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