Matrices & Determinants
Matrices & Determinants
star_batch_jee_advanced_2025
Grade None
Question:
Let $A = [a_{ij}]_{3 \times 3}$, $B = [b_{ij}]_{3 \times 3}$ where $b_{ij} = 3^{i-j}a_{ij}$, $C = [c_{ij}]_{3 \times 3}$, where $c_{ij} = 4^{i-j}b_{ij}$ be any three matrices. If $|A| = 4$, then $|B| + |C| = (|X|$ denotes determinant of matrix $X)$
Step-by-Step Solution
Key Concept: Elementary column operations (scaling columns) affect the determinant by the product of scaling factors, but carefully chosen operations preserve it.
For matrix $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$ and $B = \begin{pmatrix} a_{11} & \frac{1}{3}a_{12} & \frac{1}{9}a_{13} \\ 3a_{21} & a_{22} & \frac{1}{3}a_{23} \\ 9a_{31} & 3a_{32} & a_{33} \end{pmatrix}$, the determinant $|B| = |A|$ because $B$ is obtained by column operations that preserve the determinant. Similarly, $|C| = |B|$.
Correct Answer: 1