Show the matrix - l1 m1 n1- l2 m2 n2- l3 m3 n3 - is orthogonal if l21-m21 - n21 - - l12 - 1 - - l22 - - l32 and l1l2 - m1m2 - n1n2 - -l1l2 - 0 - -l2l3 - -l3l1

Let A = [ l_1 amp; m_1 amp; n_1; #160;l_2 amp; m_2 amp; n_2; #160;l_3 amp; m_3 amp; n_3 ] ∴ A' = [ l_1 amp; l_2 amp; l_3; m_1 amp; m_ ...

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Null Matrix

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Question

Show the matrix $⎡ ⎢ ⎣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ⎤ ⎥ ⎦$ is orthogonal if
$l_{1}^{2} + m_{1}^{2} + n_{1}^{2} = \sum l_{1}^{2} = 1 = \sum l_{2}^{2} = \sum l_{3}^{2}$ and $l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = \sum l_{1} l_{2} = 0 = \sum l_{2} l_{3} = \sum l_{3} l_{1}$

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l_{1}, m_{1}, n_{1}; l_{2}, m_{2}, n_{2}; l_{3}, m_{3}, n_{3}

l_{1}^{2} + m_{1}^{2} + n_{1}^{2} = l_{2}^{2} + m_{2}^{2} + n_{2}^{2} = l_{3}^{2} + m_{3}^{2} + n_{3}^{2} = 1

l_{2} l_{3} + m_{2} m_{3} + n_{2} n_{3} = l_{3} l_{1} + m_{3} m_{1} + n_{3} n_{1} = l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0

∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣ = \pm k

k

l_{1}^{2} + m_{1}^{2} + n_{1}^{2} = 1

l_{2}^{2} + m_{2}^{2} + n_{2}^{2} = 1

l_{3}^{2} + m_{3}^{2} + n_{3}^{2} = 1

l_{3} l_{1} + m_{3} m_{1} + n_{2} n_{1} = 0

∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{2} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣

(l_{1}, m_{1}, n_{1})

(l_{2}, m_{2}, n_{2})

l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0

l_{1}^{2} + m_{1}^{2} + n_{1}^{2} = 1, e t c a n d l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0

Δ = ∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣

l_{i}^{2} + m_{i}^{2} + n_{i}^{2} = 1

l_{i} l_{j} + m_{i} m_{j} + n_{i} n_{j} = 0

i \neq j

i, j \in {1, 2, 3}

A = ⎡ ⎢ ⎣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ⎤ ⎥ ⎦, then | A | =

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