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Consider the following system of linear equations : $$2x_1 - x_2 + 3x_3 = 1$$ $$3x_1 + 2x_2 + 5x_3 = 2$$ $$-x_1+4x_2+x_3 = 3$$ The system of equations has

1. no solution
2. a unique solution
3. more than one but a finite number of solutions
4. an infinite number of solutions
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This might help ...

rank of matrix $=$ rank of augmented matrix $=$ no of unknown $=$ $3$
so unique solution..
edited
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Can case C arise? If Yes, how shall we determine?
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when rank of matrix = rank of augmented matrixno of unknown

+2
then it is infinite solutions. r < n, that is option D. I'm asking about option C
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i think more than one but a finite number of solutions will never arise

as we have only 3 cases r=n,r<n and r>n
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yes c option case can never arise

Determinant of matrix =14 which is non zero

If The determinant of the coefficient matrix is non zero then definitely the system of given equation has a unique solution

so option B