Let X and Y be two arbitrary 3x3 non-zero skew-symmetric matrices and Z be an arbitrary 3x3 non-zero symmetric matrix. Then which of the following matrices is (are) skew symmetric?

The correct options are C, D.
X4Z3−Z3X4 and X23+Y23
Given: XT=−X, YT=−Y and ZT=Z
Using Properties of transpose:
(A+B)T=AT+BT and (AB)T=BTAT
Let's check each option:
Option A: (X44+Y44)T = X44T+Y44T = X44+Y44 (Since X and Y are skew-symmetric, their powers are either symmetric or skew-symmetric depending on the power. This would require detailed calculation to verify if it's skew-symmetric or not.)
Option B: (Y3Z4−Z4Y3)T = (Y3Z4)T−(Z4Y3)T = Z4TY3T−Y3TZ4T = Z4(−Y3)−(−Y3)Z4 = −Z4Y3+Y3Z4 = −(Y3Z4−Z4Y3) (Skew-symmetric)
Option C: (X4Z3−Z3X4)T = (X4Z3)T−(Z3X4)T = Z3TX4T−X4TZ3T = Z3(−X4)−(−X4)Z3 = −Z3X4+X4Z3 = −(X4Z3−Z3X4) (Skew-symmetric)
Option D: (X23+Y23)T = X23T+Y23T = −X23−Y23 = −(X23+Y23) (Skew-symmetric)
Therefore, options C and D are skew-symmetric.