1. Tardigrade
  2. Question
  3. Mathematics
  4. If A = [ 1 2 2 2 1 -2 a 2 b ] is a matrix satisfying the equation AAT =9 I, where, I is 3 × 3 identity matrix, then the ordered pair (a, b) is equal to

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Q. If A = $ \begin {bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end {bmatrix}$ is a matrix satisfying the equation $AA^T =9 I,$ where, $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to

JEE MainJEE Main 2015Matrices

Solution:

$AAT = 9\, I$
$\begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b\end{bmatrix}\begin{bmatrix}1 & 2 & a \\ 2 & 1 & 2 \\ 2 & -2 & b\end{bmatrix}=\begin{bmatrix}9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9\end{bmatrix}$
$a+4+2 \,b=0 $
$\Rightarrow a+2 b=-4\,\,\,\,\,\,\,\,...,(i)$
$2 a+2-2\, b=0$
$ \Rightarrow a-b=-1\,\,\,\,\,\,\,\,\,...(ii)$
From i and ii
$3\, b=-3 $
$\Rightarrow b=-1$
$a=-2$