Question

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 ______.

Options

• 24

• 18

• 45

• 36

Answer 1

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 36.

Explanation:

Since A = `1/2("A" + "A"^"T") + 1/2("A" - "A"^"T")`

Where A + A^T is symmetric and A – A^T is skew-symmetric matrix.

⇒ P = `1/2("A" + "A"^"T")` and Q = `1/2("A" - "A"^"T")`

⇒ Q = `1/2[(0, 3 - "a"),("a" - 3, 0)]`

⇒ det(Q) = `1/4("a" - 3)^2` = 9

⇒ (a – 3)² = 36

⇒ a = 9 or –3

Now, P = `1/2[(4, 3 + "a"),("a" + 3, 0)]`

⇒ det(P) = `(-1)/4("a" + 3)^2` = 36 or 0

⇒ So, Modulus of all possible values of det(P) = 36.