GATE Overflow | Mathematics | Test 1 | Question: 4

Question

Let  $Ax  =  b$  be  a  system  of  linear  equations  where  $A$  is  a  $m \times n$  matrix  and $b$ is a $m \times  1$  column  vector  and  $X$  is  a  $n \times 1$ column  vector  of  unknows.  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  $m  <  n$  and  $b$  is  the  zero  vector,  then  the  system  has  infinitely  many  solutions.
• C. If $m = n$ and $b$ is 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 1

"If m=n and b is non-zero vector, then the system has a unique solution" is FALSE actuallty.
"If m=n and b is non-zero vector AND rank of matrix is also m, then the system has a unique solution" is TRUE.

Answer 2

c is correct as when m=n the matrix weill have unique soln.