If li2 -mi2 -ni2 -1 for i -1-2-3 li lj -mi mj -ni nj -0 for i-j -1-2-3 and i - j and - - l1 m1 n1 l2 m2 n2 l3 m3 n3- then

Δ^2 = l_1 amp; m_1 amp; n_1 nbsp; l_2 amp; m_2 amp; n_2 nbsp; l_3 amp; m_3 amp; n_3 l_1 amp; l_2 amp; l_3 nbsp; m_1 amp; m ...

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Byju's Answer

Standard XII

Mathematics

Evaluation of a Determinant

If li2 +mi2 +...

Question

If $l_{i}^{2} + m_{i}^{2} + n_{i}^{2} = 1$ for $i = 1, 2, 3$ & $l_{i} l_{j} + m_{i} m_{j} + n_{i} n_{j} = 0$ for $i, j \in {1, 2, 3}$ and $i \neq j$ and $Δ = ∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣,$ then

| Δ | = 3

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| Δ | = 2

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| Δ | = 1

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| Δ | = 0

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Solution

The correct option is C $| Δ | = 1$
$Δ^{2} = ∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} l_{1} & l_{2} & l_{3} m_{1} & m_{2} & m_{3} n_{1} & n_{2} & n_{3} \end{matrix} ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} l_{1}^{2} + m_{1}^{2} + n_{1}^{2} & l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} & l_{1} l_{3} + m_{1} m_{3} + n_{1} n_{3} l_{2} l_{1} + m_{2} m_{1} + n_{2} n_{1} & l_{2}^{2} + m_{2}^{2} + n_{2}^{2} & l_{2} l_{3} + m_{2} m_{3} + n_{2} n_{3} l_{3} l_{1} + m_{3} m_{1} + n_{3} n_{1} & l_{3} l_{2} + m_{3} m_{2} + n_{3} n_{2} & l_{3}^{2} + m_{3}^{2} + n_{3}^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
$Δ^{2} = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ∣ ∣ ∣ ∣ = 1$
$\Rightarrow | Δ | = \pm 1$

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Similar questions

Q. If

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

for

i = 1, 2, 3

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

for

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

and

i \neq j

and

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

then

Q. 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 | =

Q. In a third order matrix

A, a_{i j}

denotes the element in the

i - th

row and

j - th

column.
If

a_{i j} = 0

for

i = j

= 1

for

i > j

= - 1

for

i < j

Then the matrix is

Q. In a third order matrix

A

a_{i j}

denotes the element in the i-th row and j-th column.
If

a_{i j} = 0

for

i = j

= 1

for

i > j

= - 1

for

i < j

Then the matrix is

Q. Let

Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} a_{11} & a_{12} & a_{13} a_{21} & a_{22} & a_{23} a_{31} & a_{32} & a_{33} \end{matrix} ∣ ∣ ∣ ∣,

Δ_{1} \neq 0

Δ_{2} = ∣ ∣ ∣ ∣ \begin{matrix} b_{11} & b_{12} & b_{13} b_{21} & b_{22} & b_{23} b_{31} & b_{32} & b_{33} \end{matrix} ∣ ∣ ∣ ∣

where

b_{i j}

is cofactor of

a_{i j}

\forall i,

j = 1, 2, 3

and

Δ_{3} = ∣ ∣ ∣ ∣ \begin{matrix} c_{11} & c_{12} & c_{13} c_{21} & c_{22} & c_{23} c_{31} & c_{32} & c_{33} \end{matrix} ∣ ∣ ∣ ∣

where

c_{i j}

is cofactor of

b_{i j}

\forall i,

j = 1, 2, 3,

then which one of the following is always correct.

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If l2i+m2i+n2i=1 for i=1,2,3 & lilj+mimj+ninj=0 for i,j∈{1,2,3} and i≠j and Δ=∣∣ ∣∣l1m1n1l2m2n2l3m3n3∣∣ ∣∣, then

If $l_{i}^{2} + m_{i}^{2} + n_{i}^{2} = 1$ for $i = 1, 2, 3$ & $l_{i} l_{j} + m_{i} m_{j} + n_{i} n_{j} = 0$ for $i, j \in {1, 2, 3}$ and $i \neq j$ and $Δ = ∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣,$ then