For any 3 × 3 matrix M , let | M | denote the determinant of M . Let E = 1 2 3 2 3 4 8 13 18 ,    P = 1 0 0 0 0 1 0 1 0   &   F = 1 3 2 8 18 13 2 4 3 If Q is a nonsingular matrix of order 3 × 3 , then which of the following statements is (are) TRUE ?
D. Sum of the diagonal entries of P - 1 E P + F is equal to the sum of diagonal entries of E + P - 1 F P
(1) P 2 = 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 = 1 0 0 0 1 0 0 0 1 = I PEP = 1 0 0 0 0 1 0 1 0 1 2 3 2 3 4 8 13 18 1 0 0 0 0 1 0 1 0 = 1 2 3 8 13 18 2 3 4 1 0 0 0 0 1 0 1 0 = 1 3 2 8 18 13 2 4 3 = F (2) Here, F = P E P ⇒ P F = P 2 E P ⇒ P F = E P , E = 0 ,   F = 0 E Q + P F Q - 1 = E Q + E P Q - 1 = ∣ E Q + P Q - 1 | = | E | | Q + P Q - 1 ∣ = 0 | E Q | + P F Q - 1 = 0 + 0 = 0 (3) Since, | E | = 0 ⇒ | E F | = 0 Thus, E F 3 > E F 2 is not possible (4) P = 1 0 0 0 0 1 0 1 0 ⇒ P = - 1 P - 1 =   A d j   P | P | = - - 1 0 0 0 0 - 1 0 - 1 0 = 1 0 0 0 0 1 0 1 0 P - 1 E P = 1 0 0 0 0 1 0 1 0 1 3 2 2 4 3 8 18 13 = 1 3 2 8 18 13 2 4 3 = F F + P - 1 E P = F + F = 2 F = 2 6 4 16 36 26 4 8 6 Sum of Diagonal elements of F + P - 1 E P = 44 E + P - 1 F P = E + 1 0 0 0 0 1 0 1 0 1 3 2 8 18 13 2 4 3 1 0 0 0 0 1 0 1 0 = 1 2 3 2 3 4 8 13 18 + 1 0 0 0 0 1 0 1 0 1 2 3 8 13 18 2 3 4 = 1 2 3 2 3 4 8 13 18 + 1 2 3 2 3 4 8 13 18 = 2 4 6 4 6 8 16 26 36 Tr E + P - 1 F P = 44
Related: Mathematics — Matrices · All PYQ Banks