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

Question

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

Tardigrade Admin · Accepted Answer

AAT = 9 I [1 2 2 2 1 -2 a 2 b][1 2 a 2 1 2 2 -2 b]=[9 0 0 0 9 0 0 0 9] a+4+2 b=0 ⇒ a+2 b=-4 ...,(i) 2 a+2-2 b=0 ⇒ a-b=-1 ...(ii) From i and ii 3 b=-3 ⇒ b=-1 a=-2

Q. If A = $⎣ ⎡ 12 a 212 2 - 2 b ⎦ ⎤$ is a matrix satisfying the equation $A A^{T} = 9 I,$ where, $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to

(2,-1)

(-2,1)

(2,1)

(-2,-1)

$AA T = 9 I$
$⎣ ⎡ 12 a 212 2 - 2 b ⎦ ⎤ ⎣ ⎡ 122 21 - 2 a 2 b ⎦ ⎤ = ⎣ ⎡ 900090009 ⎦ ⎤$
$a + 4 + 2 b = 0$
$\Rightarrow a + 2 b = - 4 ..., (i)$
$2 a + 2 - 2 b = 0$
$\Rightarrow a - b = - 1 ... (ii)$
From i and ii
$3 b = - 3$
$\Rightarrow b = - 1$
$a = - 2$

Q. If A = ⎣⎡​12a​212​2−2b​⎦⎤​ is a matrix satisfying the equation AAT=9I, where, I is 3×3 identity matrix, then the ordered pair (a,b) is equal to