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For 3 x 3 matrices M and N, which of the following statement(s) is (are) NOT correct ?
- a)NTMN 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(MN) for all invertible matrices M and N
Correct answer is option 'C,D'. Can you explain this answer?
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For 3 x 3 matrices M and N, which of the following statement(s) is (ar...
(A) (NTMN)T = NTMTN = NTMN if M is symmetric and is – NTMN if M is skew symmetric
(B) (MN - NM)T = NTMT - MTNT = NM - MN = –(MN – NM). So, (MN – NM) is skew symmetric
(C) (MN)T = NTMT = NM
MN if M and N are symmetric. So, MN is not symmetric
(D) (adj. M) (adj. N) = adj(NM) adj (MN).
(B) (MN - NM)T = NTMT - MTNT = NM - MN = –(MN – NM). So, (MN – NM) is skew symmetric
(C) (MN)T = NTMT = NM
MN if M and N are symmetric. So, MN is not symmetric(D) (adj. M) (adj. N) = adj(NM)
adj (MN).Most Upvoted Answer
For 3 x 3 matrices M and N, which of the following statement(s) is (ar...
Statement A: NTMN is symmetric or skew symmetric, according as M is symmetric or skew symmetric
To prove this statement, let's consider two cases:
Case 1: M is symmetric
In this case, M = M^T. Now, let's calculate NTMN:
NTMN = (N^T)(M)(N)
= (N^T)(M^T)(N) [Since M = M^T]
= (N^T)(N^T)(M) [Using the property (AB)^T = B^TA^T]
= (N^T)(N)(M) [Since N = N^T]
= (NN^T)(M) [Using the property (AB)^T = B^TA^T]
= (N^2)(M) [Since N = N^T]
= (NN)(M) [Since N = N^T]
We can see that (NN)(M) is equal to (N^2)(M), which means NTMN is symmetric.
Case 2: M is skew symmetric
In this case, M = -M^T. Now, let's calculate NTMN:
NTMN = (N^T)(M)(N)
= (N^T)(-M^T)(N) [Since M = -M^T]
= -(N^T)(M^T)(N) [Using the property (AB)^T = B^TA^T]
= -(N^T)(-N^T)(M) [Using the property (AB)^T = B^TA^T]
= -(N^T)(-N)(M) [Since N = N^T]
= (NN)(M) [Since negative signs cancel out]
We can see that (NN)(M) is equal to -(NN)(M), which means NTMN is skew symmetric.
Therefore, statement A is correct.
Statement B: MN - NM is skew symmetric for all symmetric matrices M and N
To prove this statement, let's consider two symmetric matrices M and N:
MN - NM = (M)(N) - (N)(M)
= (MN) - (NM)
Now, let's calculate the transpose of (MN - NM):
[(MN - NM)^T] = [(MN)^T - (NM)^T]
= [N^T M^T - M^T N^T]
= [NM - MN]
We can see that [(MN - NM)^T] is equal to [NM - MN], which means MN - NM is skew symmetric.
Therefore, statement B is correct.
Statement C: MN is symmetric for all symmetric matrices M and N
To disprove this statement, we can provide a counterexample:
Let M = [1 2 3] and N = [4 5 6]
[2 3 4] [5 6 7]
[3 4 5] [6 7 8]
M is symmetric and N is symmetric. However, MN = [32 38 44], which is not symmetric.
Therefore, statement C is incorrect.
Statement D: (adj M)(adj N) = adj
To prove this statement, let's consider two cases:
Case 1: M is symmetric
In this case, M = M^T. Now, let's calculate NTMN:
NTMN = (N^T)(M)(N)
= (N^T)(M^T)(N) [Since M = M^T]
= (N^T)(N^T)(M) [Using the property (AB)^T = B^TA^T]
= (N^T)(N)(M) [Since N = N^T]
= (NN^T)(M) [Using the property (AB)^T = B^TA^T]
= (N^2)(M) [Since N = N^T]
= (NN)(M) [Since N = N^T]
We can see that (NN)(M) is equal to (N^2)(M), which means NTMN is symmetric.
Case 2: M is skew symmetric
In this case, M = -M^T. Now, let's calculate NTMN:
NTMN = (N^T)(M)(N)
= (N^T)(-M^T)(N) [Since M = -M^T]
= -(N^T)(M^T)(N) [Using the property (AB)^T = B^TA^T]
= -(N^T)(-N^T)(M) [Using the property (AB)^T = B^TA^T]
= -(N^T)(-N)(M) [Since N = N^T]
= (NN)(M) [Since negative signs cancel out]
We can see that (NN)(M) is equal to -(NN)(M), which means NTMN is skew symmetric.
Therefore, statement A is correct.
Statement B: MN - NM is skew symmetric for all symmetric matrices M and N
To prove this statement, let's consider two symmetric matrices M and N:
MN - NM = (M)(N) - (N)(M)
= (MN) - (NM)
Now, let's calculate the transpose of (MN - NM):
[(MN - NM)^T] = [(MN)^T - (NM)^T]
= [N^T M^T - M^T N^T]
= [NM - MN]
We can see that [(MN - NM)^T] is equal to [NM - MN], which means MN - NM is skew symmetric.
Therefore, statement B is correct.
Statement C: MN is symmetric for all symmetric matrices M and N
To disprove this statement, we can provide a counterexample:
Let M = [1 2 3] and N = [4 5 6]
[2 3 4] [5 6 7]
[3 4 5] [6 7 8]
M is symmetric and N is symmetric. However, MN = [32 38 44], which is not symmetric.
Therefore, statement C is incorrect.
Statement D: (adj M)(adj N) = adj
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For 3 x 3 matrices M and N, which of the following statement(s) is (are) NOT correct ?a)NTMN is symmetric or skew symmetric, according as M is symmetric or skew symmetricb)MN - NM is skew symmetric for all symmetric matrices M and Nc)MN is symmetric for all symmetric matrices M and Nd)(adj M) (adj N) = adj(MN) for all invertible matrices M and NCorrect answer is option 'C,D'. Can you explain this answer?
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For 3 x 3 matrices M and N, which of the following statement(s) is (are) NOT correct ?a)NTMN is symmetric or skew symmetric, according as M is symmetric or skew symmetricb)MN - NM is skew symmetric for all symmetric matrices M and Nc)MN is symmetric for all symmetric matrices M and Nd)(adj M) (adj N) = adj(MN) for all invertible matrices M and NCorrect answer is option 'C,D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about For 3 x 3 matrices M and N, which of the following statement(s) is (are) NOT correct ?a)NTMN is symmetric or skew symmetric, according as M is symmetric or skew symmetricb)MN - NM is skew symmetric for all symmetric matrices M and Nc)MN is symmetric for all symmetric matrices M and Nd)(adj M) (adj N) = adj(MN) for all invertible matrices M and NCorrect answer is option 'C,D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For 3 x 3 matrices M and N, which of the following statement(s) is (are) NOT correct ?a)NTMN is symmetric or skew symmetric, according as M is symmetric or skew symmetricb)MN - NM is skew symmetric for all symmetric matrices M and Nc)MN is symmetric for all symmetric matrices M and Nd)(adj M) (adj N) = adj(MN) for all invertible matrices M and NCorrect answer is option 'C,D'. Can you explain this answer?.
For 3 x 3 matrices M and N, which of the following statement(s) is (are) NOT correct ?a)NTMN is symmetric or skew symmetric, according as M is symmetric or skew symmetricb)MN - NM is skew symmetric for all symmetric matrices M and Nc)MN is symmetric for all symmetric matrices M and Nd)(adj M) (adj N) = adj(MN) for all invertible matrices M and NCorrect answer is option 'C,D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about For 3 x 3 matrices M and N, which of the following statement(s) is (are) NOT correct ?a)NTMN is symmetric or skew symmetric, according as M is symmetric or skew symmetricb)MN - NM is skew symmetric for all symmetric matrices M and Nc)MN is symmetric for all symmetric matrices M and Nd)(adj M) (adj N) = adj(MN) for all invertible matrices M and NCorrect answer is option 'C,D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For 3 x 3 matrices M and N, which of the following statement(s) is (are) NOT correct ?a)NTMN is symmetric or skew symmetric, according as M is symmetric or skew symmetricb)MN - NM is skew symmetric for all symmetric matrices M and Nc)MN is symmetric for all symmetric matrices M and Nd)(adj M) (adj N) = adj(MN) for all invertible matrices M and NCorrect answer is option 'C,D'. Can you explain this answer?.
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ample number of questions to practice For 3 x 3 matrices M and N, which of the following statement(s) is (are) NOT correct ?a)NTMN is symmetric or skew symmetric, according as M is symmetric or skew symmetricb)MN - NM is skew symmetric for all symmetric matrices M and Nc)MN is symmetric for all symmetric matrices M and Nd)(adj M) (adj N) = adj(MN) for all invertible matrices M and NCorrect answer is option 'C,D'. Can you explain this answer? tests, examples and also practice JEE tests.
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