Let X and' be two arbltrary, 3 x 3. non-zero. skew-symmetric ma'u\c.es and Z be an arbitrary 3 x 3, non-zero, symmetric matrix. Then which of the following matrices is (are) skew-symmetric?
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Answer:
Given: X
T
=−X,Y
T
=−Y and Z
T
=Z
Using Properties of transpose:
(A+B)
T
=A
T
+B
T
and (AB)
T
=B
T
A
T
Option A:
(Y
3
Z
4
−Z
4
Y
3
)
T
=(Y
3
Z
4
)
T
−(Z
4
Y
3
)
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
=Y
3
Z
4
−Z
4
Y
3
Its a symmetric.
Option B:
(X
44
+Y
44
)
T
=(X
T
)
44
+(Y
T
)
44
=X
44
+Y
44
Its a symmetric.
Option C:
(X
4
Z
3
−Z
3
X
4
)
T
=(X
4
Z
3
)
T
−(Z
3
X
4
)
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
=−(X
4
Z
3
−Z
3
X
4
)
Its a skew symmetric.
Option D:
(X
23
+Y
23
)
T
=(X
T
)
23
+(Y
T
)
23
=−(X
23
+Y
23
)
Its a skew symmetric.
Hence, option C,D.
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