Matrices & Determinants
Matrices & Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
System of equations $ax + 4y + z = 0, 2y + 3z = 1, 3x - bz = -2$ then which of the following is not true
Unique solution if ab ≠ 15
Infinitely many solutions if a = 3, b = 5
No solution if ab = 15, a ≠ 3
No solution if ab = 15, a ≠ 5
Step-by-Step Solution
Key Concept: The system has a unique solution if $|A| \neq 0$; for non-unique solutions, the determinant must equal zero.
For the system $ax + 4y + z = 0$; $2y + 3z = -1$; $3x - bz = -2$, the coefficient matrix has determinant $|A| = \begin{vmatrix} a & 4 & 1 \\ 0 & 2 & 3 \\ 3 & 0 & -b \end{vmatrix} = a(2(-b) - 0) - 4(0 - 9) + 1(0-6) = -2ab + 30 - 6$. Setting $|A| = 0$ gives $-2ab + 30 = 0$, or relationships between $a$ and $b$ for consistency.
Correct Answer: 4