Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
If $a, b$ and $c$ are distinct positive real numbers such that $\Delta_1 = \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$ and $\Delta_3 = \begin{vmatrix} bc - a^2 & ac - b^2 & ab - c^2 \\ ac - b^2 & ab - c^2 & bc - a^2 \\ ab - c^2 & bc - a^2 & ac - b^2 \end{vmatrix}$, then
$\Delta_1 = \Delta_3$
$\Delta_1^2 = \Delta_3$
$\Delta_1^2 + \Delta_3 = 0$
$\Delta_1 = \Delta_3^2$
Step-by-Step Solution
Key Concept: The relationship between a determinant and the determinant of its cofactor matrix involves the square of the original determinant.
Elements of $\Delta_2$ are co-factors of the elements of $\Delta_1$. Hence $\Delta_1^2 = \Delta_2$ follows from the property that when each element is replaced by its co-factor and the determinant is computed, the result equals the square of the original determinant for certain matrix configurations.
Correct Answer: 1