Matrices & Determinants
Matrices and Determinants
Allen Star Batch
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: Recognize that the given conditions define an orthonormal set of column vectors. The matrix A with columns (a_i, b_i, c_i) satisfies A^T·A = I, making it an orthogonal matrix where det(A)² = 1, so det(A) = ±1. Given the constraint structure, the determinant equals ±1.
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