Question

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

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

Answer 1

Answer: (D)

(-2,-1)

Answer 2

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