Let a , b be non zero real numbers such that a b = 5 / 2 and given A = a - b b a and A A T = 20 I ( I is a unit matrix), then the equation whose roots are a and b is

Step-by-step solution

Given that, A = a - b b a and a b = 5 2   . . . . ( i ) ⇒ A T = a b - b a ⇒ A A T = a 2 + b 2 0 0 a 2 + b 2 Also given A A T = 20 I ⇒ a 2 + b 2 0 0 a 2 + b 2 = 20 0 0 20 ⇒ a 2 + b 2 = 20 ⇒ a + b 2 - 2 a b = 20 ⇒ a + b 2 - 2 × 5 2 = 20 [from i ] ⇒ a + b 2 = 25 ⇒ a + b = ± 5 Now, we know that general form of a quadratic equation is x 2 - sum   of   roots x + product   of   roots = 0 ⇒ x 2 ± 5 x + 5 2 = 0 ⇒ 2 x 2 ±