If YY is skew-symmetric then Y = -Y^topY=−Y⊤
If ZZ is symmetric then Z = Z^topZ=Z⊤
a)
-(Y^3Z^4-Z^4Y^3)^top =-(Z^4(Y^top)^3-(Y^top)^3Z^4) = Z^4Y^3-Y^3Z^4 = -(Y^3Z^4-Z^4Y^3)−(Y3Z4−Z4Y3)⊤=−(Z4(Y⊤)3−(Y⊤)3Z4)=Z4Y3−Y3Z4=−(Y3Z4−Z4Y3) hence a) is a symmetric matrix.
b)
X^44+Y^44 = (-X cdot X^top)^22 + (-Y cdot Y)^22X44+Y44=(−X⋅X⊤)22+(−Y⋅Y)22 which is a symmetric matrix.
c)
X^4Z^3-Z^3 X^4 = (-X X^top)^2 Z^3-Z^3 (-X X^dot)^2 = (X X^top)^2 Z^3-Z^3 (X X^top)^2 = -((X X^top)^2 Z^3-Z^3 (X X^top)^2)^topX4Z3−Z3X4=(−XX⊤)2Z3−Z3(−XX.)2=(XX⊤)2Z3−Z3(XX⊤)2=−((XX⊤)2Z3−Z3(XX⊤)2)⊤ hence this is a skew-symmetric matrix
d)
X^23+Y^23 = (-X^top)^23 + (-Y^top)^23 = -(X^top)^23-(Y^top)^23 = -(X^23+Y^23)^topX23+Y23=(−X⊤)23+(−Y⊤)23=−(X⊤)23−(Y⊤)23=−(X23+Y23)⊤ hence d) is skew-symmetric.
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