GATE | GATE CS 1996 | Question 7

Let Ax=b be a system of linear equations where A is an m×n matrix and b is a m×1 column vector and X is an n×1 column vector of unknowns. Which of the following is false?
(A) The system has a solution if and only if, both A and the augmented matrix [Ab] have the same rank
(B) If m(C) If m=n and b is a non-zero vector, then the system has a unique solution
(D) The system will have only a trivial solution when m=n, b is the zero vector and rank(A) = n


Answer: (C)

Explanation:

Following are the possibilities for a system of linear equations:

(i)  If matrix A and augmented matrix [AB]  have same  rank, then the system has solutions otherwise there is no solution.

(ii) If matrix A and augmented matrix [AB]  have same  rank which is equal to the no. of variables, then the system has unique  solutions and if B is zero vector then the system have only a trivial solution. 

(iii) If matrix A and augmented matrix [AB]  have same  rank which is less than the number of variables, then the system has infinite solutions. 

Therefore, option (C) is false because if m=n and B is non-zero vector, then it is not necessary that system has a unique solutions , because m is the number of equations ( quantity ) and not the number of linearly independent equations ( quality ).

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