Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
If $a_i^2 + b_i^2 + c_i^2 = 1, (i = 1, 2, 3)$ and $a_i a_j + b_i b_j + c_i c_j = 0$ ($i \neq j; i, j = 1, 2, 3$) then the value of $$\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$$ is ____
Step-by-Step Solution
Key Concept: The product of a matrix and its transpose determinant equals the determinant of their product.
The product of the two determinants can be written as the determinant of a matrix product. Computing $\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$ yields a matrix whose determinant is the identity matrix scaled, giving a value of $1$.
Correct Answer: 1