# 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**(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 ).

Attention reader! Don’t stop learning now. Practice GATE exam well before the actual exam with the subject-wise and overall quizzes available in **GATE Test Series Course**.

Learn all **GATE CS concepts with Free Live Classes** on our youtube channel.