Question

If $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right]$ is a matrix satisfying the equation $A{A}^T=9I$, where $I$ is $3\times 3$ identity matrix, then the ordered pair $(a,b)$ is equal :

• A. $(2,-11)$
• B. $(-2,1)$
• C. $(2,1)$
• D. $(-2,-1)$

Answer 1

The correct option is D $(-2,-1)$

$A{A}^T=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right]\left[\begin{array}{ccc}1& 2& a\\ 2& 1& 2\\ 2& -2& b\end{array}\right]=\left[\begin{array}{ccc}9& 0& a+2b+4\\ 0& 9& 2a+2-2b\\ a+2b+4& 2a+2-2b& a^2+4+b^2\end{array}\right]$
Now using given condition, $A{A}^T=9I$

$\Rightarrow \left[\begin{array}{ccc}9& 0& a+2b+4\\ 0& 9& 2a+2-2b\\ a+2b+4& 2a+2-2b& a^2+4+b^2\end{array}\right]=9\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$Therefore,$
$a+2b+4=0...\left(1\right)$
and $2a+2-2b=0\Rightarrow a-b+1=\mathrm{0...}\left(2\right)$
Solving (1) and (2) we get required ordered pair of $(a,b)\equiv (-2,-1)$