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Joint Entrance Examination
Mathematics
Matrices and Determinants
Question 299
Question 299 of 472
Medium
MCQ
Let
A
A
A
and
B
B
B
be any two
3
×
3
3\times3
3
×
3
matrices. If
A
A
A
is symmetric and
B
B
B
is skew-symmetric, then the matrix
A
B
−
B
A
AB-BA
A
B
−
B
A
is:
A
skew-symmetric
B
symmetric
C
neither symmetric nor skew-symmetric
D
I
I
I
or
−
I
-I
−
I
, where
I
I
I
is an identity matrix
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Official solution
(
A
B
−
B
A
)
T
=
(
A
B
)
T
−
(
B
A
)
T
=
B
T
A
T
−
A
T
B
T
=
(
−
B
)
(
A
)
−
(
A
)
(
−
B
)
=
−
B
A
+
A
B
=
A
B
−
B
A
(AB-BA)^T=(AB)^T-(BA)^T=B^TA^T-A^TB^T=(-B)(A)-(A)(-B)=-BA+AB=AB-BA
(
A
B
−
B
A
)
T
=
(
A
B
)
T
−
(
B
A
)
T
=
B
T
A
T
−
A
T
B
T
=
(
−
B
)
(
A
)
−
(
A
)
(
−
B
)
=
−
B
A
+
A
B
=
A
B
−
B
A
. So
A
B
−
B
A
AB-BA
A
B
−
B
A
is symmetric.
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