if lr 2 mr 2 nr 2 1r 123 and lrls mrms nrns 0r ne sr 123s

If $$l_r^2 + m_r^2 + n_r^2 = 1;r = 1,2,3$$ and $${l_r}{l_s} + {m_r}{m_s} + {n_r}{n_s} = 0;r \ne s,r = 1,2,3;s = 1,2,3,$$ then the value of \[\left| {\begin{array}{*{20}{c}} {{l_1}}&{{m_1}}&{{n_1}}\\ {{l_2}}&{{m_2}}&{{n_2}}\\ {{l_3}}&{{m_3}}&{{n_3}} \end{array}} \right|\] is

A. $$0$$

B. $$ \pm 1$$

C. $$2$$

D. None of these

Answer : $$ \pm 1$$

Solution :
\[\begin{array}{l} {D^2} = \left| {\begin{array}{*{20}{c}} {{l_1}}&{{m_1}}&{{n_1}}\\ {{l_2}}&{{m_2}}&{{n_2}}\\ {{l_3}}&{{m_3}}&{{n_3}} \end{array}} \right|\,\,\left| {\begin{array}{*{20}{c}} {{l_1}}&{{m_1}}&{{n_1}}\\ {{l_2}}&{{m_2}}&{{n_2}}\\ {{l_3}}&{{m_3}}&{{n_3}} \end{array}} \right|\\ = \,\left| {\begin{array}{*{20}{c}} {{\rm{ }}l_1^2 + m_1^2 + n_1^2}&{{l_2}{l_1} + {m_2}{m_1} + {n_2}{n_1}}&{{l_1}{l_3} + {m_1}{m_3} + {n_1}{n_3}}\\ {{\rm{ }}{l_2}{l_1} + {m_2}{m_1} + {n_2}{n_1}}&{l_2^2 + m_2^2 + n_2^2}&{{l_2}{l_1} + {m_2}{m_3} + {n_2}{n_3}}\\ {{l_1}{l_3} + {m_1}{m_3} + {n_1}{n_3}}&{{\rm{ }}{l_2}{l_3} + {m_2}{m_3} + {n_2}{n_3}}&{l_3^2 + m_3^2 + n_3^2} \end{array}} \right|\\ = \,\left| {\begin{array}{*{20}{c}} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right| = 1 \end{array}\]
$$ \Rightarrow \,D = \pm 1$$

Releted MCQ Question on
Algebra >> Matrices and Determinants

Releted Question 1

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

A. $$C$$ is empty

B. $$B$$ has as many elements as $$C$$

C. $$A = B \cup C$$

D. $$B$$ has twice as many elements as elements as $$C$$

View Answer

Releted Question 2

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

A. 0

B. 1

C. $$i$$

D. $$\omega $$

View Answer

Releted Question 3

Let $$a, b, c$$ be the real numbers. Then following system of equations in $$x, y$$ and $$z$$
$$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1,$$ $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1,$$ $$ - \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$$ has

A. no solution

B. unique solution

C. infinitely many solutions

D. finitely many solutions

View Answer

Releted Question 4

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

A. $$A + B = B + A$$

B. $$A + B = A - B$$

C. $$A - B = B - A$$

D. $$AB=BA$$

View Answer

If $$l_r^2 + m_r^2 + n_r^2 = 1;r = 1,2,3$$ and $${l_r}{l_s} + {m_r}{m_s} + {n_r}{n_s} = 0;r \ne s,r = 1,2,3;s = 1,2,3,$$ then the value of \[\left| {\begin{array}{*{20}{c}} {{l_1}}&{{m_1}}&{{n_1}}\\ {{l_2}}&{{m_2}}&{{n_2}}\\ {{l_3}}&{{m_3}}&{{n_3}} \end{array}} \right|\] is

Releted MCQ Question on
Algebra >> Matrices and Determinants

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

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Matrices and Determinants

Practice More MCQ Question on Maths Section

If $$l_r^2 + m_r^2 + n_r^2 = 1;r = 1,2,3$$ and $${l_r}{l_s} + {m_r}{m_s} + {n_r}{n_s} = 0;r \ne s,r = 1,2,3;s = 1,2,3,$$ then the value of \[\left| {\begin{array}{*{20}{c}} {{l_1}}&{{m_1}}&{{n_1}}\\ {{l_2}}&{{m_2}}&{{n_2}}\\ {{l_3}}&{{m_3}}&{{n_3}} \end{array}} \right|\] is

Releted MCQ Question on Algebra >> Matrices and Determinants

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then \[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

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Practice More MCQ Question on Maths Section

Releted MCQ Question on
Algebra >> Matrices and Determinants

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

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Matrices and Determinants