Let X a n d Y be two arbitrary, 3 × 3 , non - zero, skew - symmetric matrices and Z be an arbitrary 3 × 3 , non - zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric ?

Step-by-step solution

X T = - X , Y T = - Y and Z T = Z P = Y 3 Z 4 - Z 4 Y 3 P T = Z 4 T Y 3 T - Y 3 T Z 4 T = Z T 4 Y T 3 - Y T 3 Z T 4 = Z 4 - Y 3 - - Y 3 Z 4 = Y 3 Z 4 – Z 4 Y 3 = P ( s y m m e t r i c ) Q = X 44 + Y 44 Q T = X T 44 + Y T 44 = X 44 + Y 44 = Q(symmetric) R T = Z 3 T X 4 T - X 4 T Z 3 T = Z T 3 X T 4 - X T 4 Z T 3 = Z 3 - X 4 - - X 4 Z 3 = Z 3 X 4 - X 4 Z 3 = - R (Skew Symmetric) S = X 23 + Y 23 S T = X T 23 + Y T 23 = - X 23 + - Y 23 = - X 23 - Y 23 = - S (Skew symmetric)