Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
Let $\Delta_1 = \begin{vmatrix} a & b & c \\ e & d & c+d \\ a & b & a+b+c \end{vmatrix}$ and $\Delta_2 = \begin{vmatrix} a & b & a+c \\ b & d & b+d \\ a & c & a+d+c \end{vmatrix}$ then the value of $\left|\frac{\Delta_1}{\Delta_2}\right|$, where $b \neq 0$ and $ad \neq bc$ is ____
Step-by-Step Solution
Key Concept: Strategic column operations create a common factor in the numerator and denominator to simplify the ratio of determinants.
Using the column operation $C_3 \to C_3 - (C_1 + C_2)$ on both $D_1$ and $D_2$, compute $\frac{D_1}{D_2} = \frac{-2b(ad-bc)}{b(ad-bc)} + 2 = -2 + 2$.
Correct Answer: 2