Question

Let AX = B be a system of linear equations when A is an m x n matrix B is an n x 1 column matrix which of the following is false?

• A. The system has a solution, If $\rho$(A) = $\rho$(A/B)
• B. If m = n and B is non-zero vector then till system has a unique solution
• C. If m < n and B is a zero vector then the system has infinitely many solutions
• D. The system will have a trivial solution when m = n, B is the zero vector and rank of A is n

Answer 1

The correct option is B If m = n and B is non-zero vector then till system has a unique solution

Given that $A_{mxn}{X}_{nx1}=B_{mx1}$ ......(1)

According to option (b)

We can take m = n & B = O

So (1) $\Rightarrow$ $A_{mxn}{X}_{nx1}=O_{nx1}$

If $\left|A\right|\ne 0$ , system have unique solution, if $\left|A\right|$ = 0 system have infinite solutions.

Hence, option (b) is wrong because condition of unique solution is not mentioned.