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 vectors forming the rows of matrix A. The constraint that ||v_i|| = 1 and v_i · v_j = 0 (i≠j) means A is an orthogonal matrix, satisfying A·A^T = I, which implies det(A)·det(A^T) = 1, so det(A) = ±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