JEE Main202225 Jun 2022Morning ShiftMathematicsMatricesActual
Let A = 0 - 2 2 0 . If M and N are two matrices given by M = ∑ k = 1 10 A 2 k and N = ∑ k = 1 10 A 2 k - 1 then M N 2 is
Options
- Aa non-identity symmetric matrix
- Ba skew-symmetric matrix
- Cneither symmetric nor skew-symmetric matrix
- Dan identity matrix
Correct answer
A. a non-identity symmetric matrix
Step-by-step solution
Since A = 0 - 2 2 0 So A 2 = 0 - 2 2 0 0 - 2 2 0 = - 4 0 0 - 4 = - 4 I and A 3 = - 4 A Similarly A 4 = - 4 I - 4 I = - 4 2 I , A 5 = - 4 2 A ,      A 6 = - 4 3 I Now M = ∑ k = 1 10 A 2 k = A 2 + A 4 + … . + A 20 = - 4 + - 4 2 + - 4 3 + … . + - 4 20 I = - k 1 I So M is symmetric matrix N = ∑ k = 1 10   A 2 k - 1 = A + A 3 + … … + A 19 = A 1 + - 4 + - 4 2 + … … + - 4 9 = k 2 A So N is skew symmetric ⇒ N 2 is symmetric matrix Hence, M N 2 is non