if vartriangle 1 20c 1 and 1 and 1 a and b and c a 2 and b

If \[{\vartriangle _1} = \left| {\begin{array}{*{20}{c}} 1&1&1 \\ a&b&c \\ {{a^2}}&{{b^2}}&{{c^2}} \end{array}} \right|,{\vartriangle _2} = \left| {\begin{array}{*{20}{c}} 1&{bc}&a \\ 1&{ca}&b \\ 1&{ab}&c \end{array}} \right|\] then

A. $${\vartriangle _1} + {\vartriangle _2} = 0$$

B. $${\vartriangle _1} + {2\vartriangle _2} = 0$$

C. $${\vartriangle _1} = {\vartriangle _2}$$

D. None of these

Answer : $${\vartriangle _1} + {\vartriangle _2} = 0$$

Solution :
\[{\vartriangle _2} = \frac{1}{{abc}}\left| {\begin{array}{*{20}{c}} a&{abc}&{{a^2}} \\ b&{cab}&{{b^2}} \\ c&{abc}&{{c^2}} \end{array}} \right|\]
\[{\vartriangle _2} = \left| {\begin{array}{*{20}{c}} a&1&{{a^2}} \\ b&1&{{b^2}} \\ c&1&{{c^2}} \end{array}} \right| = - \left| {\begin{array}{*{20}{c}} 1&a&{{a^2}} \\ 1&b&{{b^2}} \\ 1&c&{{c^2}} \end{array}} \right| = - {\vartriangle _1}.\]

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 \[{\vartriangle _1} = \left| {\begin{array}{{20}{c}} 1&1&1 \\ a&b&c \\ {{a^2}}&{{b^2}}&{{c^2}} \end{array}} \right|,{\vartriangle _2} = \left| {\begin{array}{{20}{c}} 1&{bc}&a \\ 1&{ca}&b \\ 1&{ab}&c \end{array}} \right|\] then

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 \[{\vartriangle _1} = \left| {\begin{array}{*{20}{c}} 1&1&1 \\ a&b&c \\ {{a^2}}&{{b^2}}&{{c^2}} \end{array}} \right|,{\vartriangle _2} = \left| {\begin{array}{*{20}{c}} 1&{bc}&a \\ 1&{ca}&b \\ 1&{ab}&c \end{array}} \right|\] then

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

If \[{\vartriangle _1} = \left| {\begin{array}{{20}{c}} 1&1&1 \\ a&b&c \\ {{a^2}}&{{b^2}}&{{c^2}} \end{array}} \right|,{\vartriangle _2} = \left| {\begin{array}{{20}{c}} 1&{bc}&a \\ 1&{ca}&b \\ 1&{ab}&c \end{array}} \right|\] then

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