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If l2i+m2i+n2...

Question

If $l_{i}^{2} + m_{i}^{2} + n_{i}^{2} = 1$ and
$l_{i} l_{j} + m_{i} m_{j} + n_{i} n_{j} = 0$ for $i \neq j$
where $i, j \in {1, 2, 3}$ and
$A = ⎡ ⎢ ⎣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ⎤ ⎥ ⎦, then | A | =$

A

4

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B

3

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C

1

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D

2

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Solution

The correct option is C $1$
$A . A^{T} = ⎡ ⎢ ⎣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} l_{1} & l_{2} & l_{3} m_{1} & m_{2} & m_{3} n_{1} & n_{2} & n_{3} \end{matrix} ⎤ ⎥ ⎦$

$= ⎡ ⎢ ⎢ ⎣ \begin{matrix} l_{1}^{2} + m_{1}^{2} + n_{1}^{2} & l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} & l_{1} l_{3} + m_{1} m_{2} + n_{1} n_{3} l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} & l_{2}^{2} + m_{2}^{2} + n_{2}^{2} & l_{2} l_{3} + m_{2} m_{3} + n_{2} n_{3} l_{1} l_{3} + m_{1} m_{2} + n_{1} n_{3} & l_{2} l_{3} + m_{2} m_{3} + n_{2} n_{3} & l_{3}^{2} + m_{3}^{2} + n_{3}^{2} \end{matrix} ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$
$∴ A . A^{T} = I \Rightarrow | A . A^{T} | = 1$
$\Rightarrow | A |^{2} = 1,$
$∵ | A | = | A^{T} |$
$\Rightarrow | A | = \pm 1$

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Similar questions

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

i = 1, 2, 3

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

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

i \neq j

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

i = 1, 2, 3

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

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

i \neq j

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

Q. 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}

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}

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

B = ⎡ ⎢ ⎣ \begin{matrix} p_{1} & q_{1} & r_{1} p_{2} & q_{2} & r_{2} p_{3} & q_{3} & r_{3} \end{matrix} ⎤ ⎥ ⎦

p_{1}, q_{1}, r_{1}

are the co-factors of the elements

l_{i}, m_{i}, n_{i}

i = 1, 2, 3

(l_{1}, m_{1}, n_{1}), (l_{2}, m_{2}, n_{2})

(l_{3}, m_{3}, n_{3})

are the direction cosines of three mutually perpendicular lines then

(p_{1}, q_{1}, r_{1}), (p_{2}, q_{2}, r_{2})

(p_{3}, q_{3}, r_{3})

Q. The value of

A_{i} A_{j}

j = 1, 2, 3, . . . n

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