If A = 1 2 2 2 1 - 2 a 2 b is a matrix satisfying the equation A A T = 9 I , where I is 3 × 3 identity matrix, then the ordered pair a , b is equal to

Step-by-step solution

Given, A A T = 9 I 1 2 2 2 1 − 2 a 2 b 1 2 a 2 1 2 2 - 2 b = 9 I ⇒ 9 0 a + 4 + 2 b 0 9 2 a + 2 − 2 b a + 4 + 2 b 2 a + 2 − 2 b a 2 + 4 + b 2 = 9 0 0 0 9 0 0 0 9 Equation a + 4 + 2 b = 0 ⇒ a + 2 b = − 4     . . . 1 2 a + 2 − 2 b = 0 ⇒ 2 a − 2 b = − 2     . . . 2 &     a 2 + 4 + b 2 = 9 ⇒ a 2 + b 2 = 5     . . . 3 On solving, we get a = − 2 ,   b = − 1