"Let a, b, c $\in$ R be all non-zero and satisfy a3 + b3 + c3 = 2. If the matrix A = $$\left( {\matrix{ a & b & c \cr b & c & a \cr c & a & b \cr } } \right)$$ satisfies ATA = I, then a value of abc can be :"

Question

Accepted Answer

"Given,
${a^3} + {b^3} + {c^3} = 2$

${A^T}A = I$

$$ \Rightarrow \left[ {\matrix{ a & b & c \cr b & c & a \cr c & a & b \cr } } \right]\left[ {\matrix{ a & b & c \cr b & c & a \cr c & a & b \cr } } \right] = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right]$$

$\Rightarrow {a^2} + {b^2} + {c^2} = 1$

and ab + bc + ca = 0

Now (a + b + c)² = 1

$\Rightarrow (a + b + c) = \pm 1$

So, ${a^3} + {b^3} + {c^3} - 3abc$

$\Rightarrow (a + b + c)\left( {{a^2} + {b^2} + {c^2} - ab - bc - ca} \right)$

$\Rightarrow \pm 1(1 - 0) = \pm 1$

$\therefore 2 - 3abc = \pm 1$

$\Rightarrow 3abc = 2 \pm 1 = 3,1$

$\Rightarrow abc = 1,{1 \over 3}$ "

Let a, b, c $\in$ R be all non-zero and satisfy
a³ + b³ + c³ = 2. If the matrix

A = $$\left( {\matrix{ a & b & c \cr b & c & a \cr c & a & b \cr } } \right)$$

satisfies A^TA = I, then a value of abc can be :

Solution

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Let a, b, c $\in$ R be all non-zero and satisfy a3 + b3 + c3 = 2. If the matrix A = $$\left( {\matrix{ a & b & c \cr b & c & a \cr c & a & b \cr } } \right)$$ satisfies ATA = I, then a value of abc can be :

Solution

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More Matrices and Determinants questions

Drill 25 more like these. Every day. Free.

Let a, b, c $\in$ R be all non-zero and satisfy
a³ + b³ + c³ = 2. If the matrix

A = $$\left( {\matrix{ a & b & c \cr b & c & a \cr c & a & b \cr } } \right)$$

satisfies A^TA = I, then a value of abc can be :