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Q.
if matrix $A = \begin{bmatrix}a&b&c\\ b&c&a\\ c&a&b\end{bmatrix}$where $a, b, c $ are real positive numbers, $abc = 1$ and $A^TA = I$, then the value of $a^3 + b^3 +c^3$ is
Matrices
Solution:
since, $A^TA =I$
$\Rightarrow \begin{bmatrix}a&b&c\\ b&c&a\\ c&a&b\end{bmatrix}\begin{bmatrix}a&b&c\\ b&c&a\\ c&a&b\end{bmatrix} = \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}$
$\Rightarrow \begin{bmatrix}a^{2}+b^{2}+c^{2}&ab+bc+ca&ab+bc+ca\\ ab+bc+ca&a^{2}+b^{2}+c^{2}&ab+bc+ca\\ ab+bc+ca&ab+bc+ca&a^{2}+b^{2}+c^{2}\end{bmatrix} = \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}$
$\Rightarrow a^2+b^2+c^2 = 1$ and $ab+bc+ca = 0$
Now, $(a+b+c)^2 = a^2 + b^2 +c^2 + 2(ab + bc + ca )$
$= 1 + 2\cdot 0 = 1$
$\Rightarrow a + b + c = 1 ..............(i)$
Now, $( a^3 +b^3 +c^3)$
$=(a+ b + c )(a^2 +b^2 +c^2 -ab - bc - ca ) +3abc $
$\Rightarrow a^3 +b^3 +c^3 = 1 + 3 = 4 \,\,[using (i)]$