Let and be two non-singular skew-symmetric matrices such that . If denotes the transpose of , then is equal to
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: C
Solution
For skew-symmetric matrices, MT = -M and NT = -N. Since M and N are 3×3 non-singular skew-symmetric matrices and MN = NM, we can simplify the expression step by step.
(MT N)-1 = (−MN)-1 = −(MN)-1 = −N-1M-1
(MN-1)T = (N-1)T MT = (NT)-1(−M) = (−N)-1(−M) = N-1M (using commutativity, this equals M N-1 ... but let's keep it as N-1M)
Now the full expression: M²N² · (−N-1M-1) · (N-1M)
= M²N² · (−N-1M-1N-1M)
Since MN = NM, M and N commute, so M-1 and N-1 also commute with everything. Thus:
−M²N² · N-1M-1N-1M = −M²N² · M-1N-2M (using commutativity to rearrange)
= −M² · (N²N-2) · (M-1M) = −M² · I · I = −M²
(MT N)-1 = (−MN)-1 = −(MN)-1 = −N-1M-1
(MN-1)T = (N-1)T MT = (NT)-1(−M) = (−N)-1(−M) = N-1M (using commutativity, this equals M N-1 ... but let's keep it as N-1M)
Now the full expression: M²N² · (−N-1M-1) · (N-1M)
= M²N² · (−N-1M-1N-1M)
Since MN = NM, M and N commute, so M-1 and N-1 also commute with everything. Thus:
−M²N² · N-1M-1N-1M = −M²N² · M-1N-2M (using commutativity to rearrange)
= −M² · (N²N-2) · (M-1M) = −M² · I · I = −M²
[AIEEE 2011]