MathematicsIf A = abcbcacab, abc = 1,AT A =I, then find the value of a3+b3+c3.

If $A = [\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}], a b c = 1, A^{T} A = I,$ then find the value of $a^{3} + b^{3} + c^{3} .$

A
3 or 5
B
5 or 6
C
7 or 6
D
2 or4

Solution:

$A^{T} A = I \Rightarrow |A^{T} A| = |I| \Rightarrow |A^{T}| |A| = 1 \Rightarrow |A| |A| = 1 [∵ |A^{T}| = |A|] \Rightarrow |A| = \pm 1 \Rightarrow |\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}| = \pm 1 \Rightarrow 3 a b c - (a^{3} + b^{3} + c^{3}) = \pm 1$

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

or $a^{3} + b^{3} + c^{3} = 3 \pm 1 = 2$ or $4$

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