Let A = 2 3 a 0 , a ∈ R be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix. If det Q = 9 , then the modulus of the sum of all possible values of determinant of P is equal to:

Step-by-step solution

We have, A = 2 3 a 0 ,   a ∈ R Then, A T = 2 a 3 0 Now, P = A + A T 2 ⇒ P = 1 2 2 3 a 0 + 2 a 3 0 ⇒ P = 2 3 + a 2 a + 3 2 0 And, Q = A - A T 2 ⇒ Q = 1 2 2 3 a 0 - 2 a 3 0 ⇒ Q = 0 3 - a 2 a - 3 2 0 As, det ( Q ) = 9 ⇒ a - 3 2 = 36 ⇒ a = 3 ± 6 ∴   a = 9 , - 3 Now, det P = - 3 + a 2 2 For a = - 3 ,   det P = 0 For a = 9 , det P = - 3 + 9 2 2 = - 36 ∴ Modulus of the sum of all possible values of det P = - 36 + 0 = 36