1
Question
Number of solutions for the system of linear equations, will be,
x + y + z =6
x + 2y + 3z = 14
x + 4y + 2z =30
Number of solutions for the system of linear equations, will be,
x + y + z =6
x + 2y + 3z = 14
x + 4y + 2z =30
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Solution
The correct option is B Infinite solution
Here A X=B⎡⎢⎣111123147⎤⎥⎦⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣61430⎤⎥⎦Let′s calculate|A||A|=1(14−12)−1(7−3)+1(4−2)=2−4+2=0∴The given system of equations either has no solution or infinite number of solutionCofactor matricx of A will be=⎡⎢⎣2−42−36−31−21⎤⎥⎦Adjoint of A will be=⎡⎢⎣2−31−46−22−31⎤⎥⎦∴adj A.B=⎡⎢⎣2−31−46−22−31⎤⎥⎦.⎡⎢⎣61430⎤⎥⎦=⎡⎢⎣000⎤⎥⎦=0.We know that if|A|=0 and adjA.B=0 then there exist infinite solutions.
Infinite solution
Here A X=B⎡⎢⎣111123147⎤⎥⎦⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣61430⎤⎥⎦Let′s calculate|A||A|=1(14−12)−1(7−3)+1(4−2)=2−4+2=0∴The given system of equations either has no solution or infinite number of solutionCofactor matricx of A will be=⎡⎢⎣2−42−36−31−21⎤⎥⎦Adjoint of A will be=⎡⎢⎣2−31−46−22−31⎤⎥⎦∴adj A.B=⎡⎢⎣2−31−46−22−31⎤⎥⎦.⎡⎢⎣61430⎤⎥⎦=⎡⎢⎣000⎤⎥⎦=0.We know that if|A|=0 and adjA.B=0 then there exist infinite solutions.
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