JEE Advanced | 2013MatricesMedium
Question
For 3 × 3 matrices M and N, which of the following statement(s) is (are) NOT correct ?
Options
A.NTM N is symmetric or skew symmetric, according as M is symmetric or skew symmetric
B. MN - NM is skew symmetric for all symmetric matrices M and N
C.MN is symmetric for all symmetric matrices M and N
D.(adj M) (adj N) = adj (M N) for all invertible matrices M and N
Solution
(A)
Let A = NT M N
AT = NT MT N
A = AT when M = MT
A = - AT when M = -MT
(B)
Let B = MN - NM & MT = M & NT = N
BT = NTMT - MTNT
BT = NM - MN
B + BT = 0
(C)
M = MT & M = NT
A = MN
A T = N TM T ⇒ A T = NM
A ≠ A T
(D)
adj(MN) = (adj N) (adj M)
Hence correct answer are C and D.
Let A = NT M N
AT = NT MT N
A = AT when M = MT
A = - AT when M = -MT
(B)
Let B = MN - NM & MT = M & NT = N
BT = NTMT - MTNT
BT = NM - MN
B + BT = 0
(C)
M = MT & M = NT
A = MN
A T = N TM T ⇒ A T = NM
A ≠ A T
(D)
adj(MN) = (adj N) (adj M)
Hence correct answer are C and D.
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