Question

If determinant of matrix is zero, then system is

• A. Unique
• B. Infinite
• C. No solution
• D. Either infinite or no solution
• E. Always consistent

Hint:

Determinant zero means system is not uniquely solvable — always check consistency.

Answer 1

The Correct Option is D

Concept: The determinant of the coefficient matrix tells us about the nature of solutions of a system of linear equations.

Step 1: Recall basic rule:
If determinant \( \neq 0 \), the system has a unique solution.

Step 2: Now consider determinant \( = 0 \).
This means the matrix is singular, so inverse does not exist.

Step 3: When inverse does not exist, we cannot directly solve uniquely.
So the system becomes dependent.

Step 4: A dependent system can behave in two ways:
• If equations are consistent → infinite solutions
• If inconsistent → no solution

Step 5: Therefore, determinant zero does not guarantee one specific case.
It can lead to either of the two situations.

Step 6: Hence final conclusion: \[ \boxed{\text{Either infinite or no solution}} \]