Question

Matrix A is called orthogonal matrix if $A{A}^T=I=A^{T}A$. Let $A=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right]$ be an orthogonal matrix. Let
$\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k},\overrightarrow{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k},\overrightarrow{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$. Then $|\overrightarrow{a}|=|\overrightarrow{b}|=|\overrightarrow{c}|=1$ & $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}=0$ i.e.
$\overrightarrow{a},\overrightarrow{b}$ & $\overrightarrow{c}$ forms mutually perpendicular triad of unit vectors.
If $abc=P$ and $Q=\left[\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right]$, where Q is an orthogonal matrix. Then.
On the basis of above information answer the following questions:The values of $a+b+c$ is -

• A. $2$
• B. $p$
• C. $2p$
• D. $\pm 1$

Answer 1

Answer: (D)

$\pm 1$

$Q=\left[\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right]$ is an orthogoanal matrix.

$\Rightarrow Q{Q}^T=I$

taking modulus on both sides gives $|Q|=\pm 1$ -----(1)

but according to given data in $Q$

$a^2+b^2+c^2=1$ and $ab+bc+ca=0$ ------(2)

but

$|Q|=\left|\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right|=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$

$\Rightarrow |Q|=a+b+c$ (from (2))

$\therefore a+b+c=\pm 1$

Hence, option D.