Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
If the system of linear equations $x + y + z = 0$, $3x + ky - 2z = 0$, $2x + 4y - 3z = 0$ has a non-zero solution $(x, y, z)$, then $\frac{xz}{y^2}$ is equal to
Step-by-Step Solution
Key Concept: Setting determinant to zero ensures the system has non-trivial solutions, which are then used to find the required ratio.
Setting the determinant equal to zero: $\begin{vmatrix} 1 & k & 3 \\ 3 & k & -2 \\ 2 & 4 & -3 \end{vmatrix} = 0$. Expanding gives $-3k + 8 - k(-9-4) + 3(12-2k) = 0$, which simplifies to $-4k = -44$, so $k = 11$. With $k=11$, the system becomes $x + 11y + 3z = 0$, $3x + 11y - 2z = 0$, $2x + 4y - 3z = 0$. Solving yields $x = \frac{5t}{2}$, $y = -\frac{t}{2}$, and $\frac{xz}{y^2} = 10$.
Correct Answer: 10