Let P 1 = I = 1 0 0 0 1 0 0 0 1 ,   P 2 = 1 0 0 0 0 1 0 1 0 ,   P 3 = 0 1 0 1 0 0 0 0 1 ,  P 4 = 0 1 0 0 0 1 1 0 0 ,   P 5 = 0 0 1 1 0 0 0 1 0 , P 6 = 0 0 1 0 1 0 1 0 0 , and X = ∑ k = 1 6 P K 2 1 3 1 0 2 3 2 1 P K T where P K T denotes the transpose of the matrix P K . Then which of the following options is/are correct?
D. X is a symmetric matrix
∵   P 1 = P 1 T = P 1 - 1 , P 3 = P 3 T = P 3 - 1 , P 5 = P 5 T = P 5 T , P 2 = P 2 T = P 2 - 1 , P 4 = P 4 T = P 4 T , P 6 = P 6 T = P 6 - 1 Now Let Q = 2 1 3 1 0 2 3 2 1 X = ∑ K = 1 6 P K Q P K T ⇒ T r a c e X = ∑ K = 1 6 T r P K Q P K T ∵ T r A B = T r B A = ∑ K = 1 6 T r P K T P K Q ∵ P K T   P K = I = ∑ K = 1 6 T r Q = 6   T r a c e Q = 6 × 3 = 18 ⇒ X = ∑ K = 1 6 P K Q P K T X T = ∑ K = 1 6 P K Q P K T T X T = ∑ K = 1 6 P K T T Q T P K T ∵ A T T = A X T = ∑ K = 1 6 P K Q P K T ∵ Q T = Q X T = X ⇒ Let R = 1 1 1 then X R = ∑ K = 1 6 P K Q P K T R X R = ∑ K = 1 6 P K Q P K R ∵ P K T = P K X R = ∑ K = 1 6 P K Q R ∵ P K R = R X R = ∑ K = 1 6 P K 6 3 6 ∵ Q R = 6 3 6 X R = 2 2 2 2 2 2 2 2 2 6 3 6 ∵   ∑ K = 1 6 P K = 2 2 2 2 2 2 2 2 2 X R = 30 30 30 ⇒ X R = 30 1 1 1 ⇒ α = 30 ⇒ X R = 30 1 1 1 X R = 30   R X - 30 I R = 0 Now x - 30 I must be zero else R = 0 If x - 30 I ≠ 0 then x - 30 I - 1 x - 30 I R = 0 R = 0 not possible Hence, x - 30 I = 0 X - 30 I is not invertible.
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