Let a- b- c- R be such that a - b - c - 0 and abc-2- Let A- a b c- b c a- c a b - If A2 - I- then value of a3 - b3 - c3 is

The correct option is D 7 A^2 = I ⇒ a^2 + b^2 + c^2=1 and bc+ca+ab=0 Also, |A^2|=1⇒ |A|=± 1 .But A|=3abc (a^3 + b^3 + c^3) = ...

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Byju's Answer

Standard XII

Mathematics

Evaluation of a Determinant

Let a, b, ...

Question

Let $a, b, c \in R$ be such that $a + b + c > 0$ and $a b c = 2$ . Let $A = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦$ If $A^{2} = I$ , then value of $a^{3} + b^{3} + c^{3}$ is

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Solution

The correct option is D 7
$A^{2} = I$
$\Rightarrow a^{2} + b^{2} + c^{2} = 1$ and $b c + c a + a b = 0$
Also, $| A^{2} | = 1 \Rightarrow | A | = \pm 1$ .
But $A | = 3 a b c - (a^{3} + b^{3} + c^{3})$
$= - (a + b + c) (a^{2} + b^{2} + c^{2} - b c - c a - a b)$
$= - (a + b + c)$
As $a + b + c > 0$ , we get $| A | = - 1$ .
$∴ a^{3} + b^{3} + c^{3} = 7$

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Similar questions

If matrix $A = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦$ where a, b, c are real positive numbers, abc = 1 and $A^{T} A = I,$ then the value of $a^{3} + b^{3} + c^{3}$ is ___

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A = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦, a b c = 1, A^{T} A = I

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a^{3} + b^{3} + c^{3}

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a^{3} + b^{3} + c^{3}

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Let a,b,c∈R be such that a+b+c>0 and abc=2. Let A=⎡⎢⎣abcbcacab⎤⎥⎦ If A2=I, then value of a3+b3+c3 is

The correct option is D 7A2=I⇒a2+b2+c2=1 and bc+ca+ab=0Also, |A2|=1⇒|A|=±1.But A|=3abc−(a3+b3+c3)=−(a+b+c)(a2+b2+c2−bc−ca−ab)=−(a+b+c)As a+b+c>0, we get |A|=−1.∴a3+b3+c3=7

Let $a, b, c \in R$ be such that $a + b + c > 0$ and $a b c = 2$ . Let $A = ⎡ ⎢ ⎣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ⎤ ⎥ ⎦$ If $A^{2} = I$ , then value of $a^{3} + b^{3} + c^{3}$ is