Question

Which of the following systems has non-trivial solution?

• A. $AX=0$, $|A|=4$
• B. $AX=0$, $|A|=-4$
• C. $AX=0$, $|A|=0$
• D. $AX=B$, $|B|=5$

Hint:

No Determinant ($|A|=0$) = No Unique Solution = Non-Trivial possibilities!

Answer 1

The Correct Option is C

Step 1: Concept
For a homogeneous system $AX=0$, a non-trivial solution exists if and only if the matrix $A$ is singular ($|A|=0$).

Step 2: Meaning
A trivial solution is $X=0$. Non-trivial solutions are non-zero vectors that satisfy the equation.

Step 3: Analysis
If $|A| \neq 0$, the system has only the unique trivial solution ($X=0$). If $|A|=0$, there are infinitely many solutions, including non-trivial ones.

Step 4: Conclusion
Option (C) matches the condition $|A|=0$ for a homogeneous system.
Final Answer: (C)