Question

If matrix $A=\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]$ where a,b,c are real numbers $A^{T}A=I$, then find the value of $a+b+c$.

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

Answer: (A), (C)

The correct options are
A $-1$
C $1$

$A^{T}A=\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\Rightarrow \left[\begin{array}{ccc}{a}^2+b^2+c^2& ab+bc+ca& ac+ba+cb\\ ab+bc+ca& b^2+c^2+a^2& bc+ca+ab\\ ac+ba+cb& bc+ac+ab& c^2+a^2+b^2\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

We know that ${(a+b+c)}^2=a^2+b^2+c^2+2(ab+bc+ca)$

But from matrix, we get $a^2+b^2+c^2=1$ & $ab+bc+ca=0$

$\therefore {(a+b+c)}^2=1+(2\times 0)\Rightarrow a+b+c=\sqrt{1}=\pm 1$