If P = a b c b c a c a b , a b c = 1 , P T P = I then the value of a 3 + b 3 + c 3 is

Given, P = a b c b c a c a b ∴ P T = a b c b c a c a b Also given P T P = I ∴ P T P = I = 1 or a b c b c a c a b 2 = 1 ⇒ a b c b c a c a b = ± 1 or 3 a b c - a 3 - b 3 - c 3 = ± 1 or 3 ∓ 1 = a 3 + b 3 + c 3 ∴ a 3 + b 3 + c 3 = 2 or 4

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MathematicsMatricesSymmetric & Skew Symmetric MatricesMedium2 minPYQ_2018

MathematicsMediumsingle choice

If $P = [\begin{array}{l} a & b & c \\ b & c & a \\ c & a & b \end{array}]$ , $a b c = 1$ , $P^{T} P = I$ then the value of $a^{3} + b^{3} + c^{3}$ is

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Answer:

Solution:

Given, $P = [\begin{array}{l} a & b & c \\ b & c & a \\ c & a & b \end{array}]$

$∴ P^{T} = [\begin{array}{l} a & b & c \\ b & c & a \\ c & a & b \end{array}]$

Also given $P^{T} P = I$

$∴ |P^{T} P| = |I| = 1$

or ${|\begin{array}{l} a & b & c \\ b & c & a \\ c & a & b \end{array}|}^{2} = 1 \Rightarrow |\begin{array}{l} a & b & c \\ b & c & a \\ c & a & b \end{array}| = \pm 1$

or $3 a b c - a^{3} - b^{3} - c^{3} = \pm 1$

or $3 \mp 1 = a^{3} + b^{3} + c^{3}$

$∴ a^{3} + b^{3} + c^{3} = 2$ or $4$

Stream:BITSATSubject:MathematicsTopic:MatricesSubtopic:Symmetric & Skew Symmetric Matrices

⏱ 2mℹ️ Source: PYQ_2018

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