Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
If $\begin{vmatrix} bc-a^2 & ca-b^2 & ab-c^2 \\ ca-b^2 & ab-c^2 & bc-a^2 \\ ab-c^2 & bc-a^2 & ca-b^2 \end{vmatrix} = \begin{vmatrix} a^2 & b^2 & b^2 \\ b^2 & a^2 & b^2 \\ b^2 & b^2 & a^2 \end{vmatrix}$, then
a^2 = a^2 + b^2 + c^2
b^2 = ab + bc + ca
a^2 = ab + bc + ca
b^2 = a^2 + b^2 + c^2
Step-by-Step Solution
Key Concept: For a cyclic matrix, the determinant of cofactors equals the square of the original determinant.
Given the matrix $\Delta = \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$, we find $(\Delta')^2 = \Delta^2$ where $\Delta'$ is the determinant of cofactors. The cofactor matrix yields $\Delta' = \begin{vmatrix} bc-a^2 & ca-b^2 & ab-c^2 \\ ca-b^2 & ab-c^2 & bc-a^2 \\ ab-c^2 & bc-a^2 & ca-b^2 \end{vmatrix}$. Computing the product shows this equals $(\Delta')^2 = \Delta^2$, confirming the relation between the original determinant and its cofactor determinant.
Correct Answer: 1,2