1. Tardigrade
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  4. If l12+m12+n12=1 etc., and l1 l2+m1 m2+n1 n2 =0 etc., and Δ=| l1 m1 n1 l2 m2 n2 l3 m3 n3 | then

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Q. If $l_1^2+m_1^2+n_1^2=1$ etc., and $l_1 l_2+m_1 m_2+n_1 n_2$ $=0$ etc., and $\Delta=\begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end{vmatrix}$ then

Determinants

Solution:

We have
$\Delta^2=\begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end{vmatrix}\begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end{vmatrix}$
$=\begin{vmatrix} a_1 & b_{21} & b_{31} \\ b_{12} & a_2 & b_{32} \\ b_{13} & b_{23} & a_3 \end{vmatrix}$
$\text { where } a_k=l_k^2+m_k^2+n_k^2=1 \text { for } k=1,2,3$
$\text { and } b_{i j}=l_i l_j+m_i m_j+n_i n_j=0 \forall i \neq j$
$\text { Thus, } \Delta^2=\begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix}=1 \Rightarrow|\Delta|=1 $