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Let A and B be the non-singular matrices of same order.
Consider the following statements:
S1:BT AB is symmetric if and only if BT = B
S2: BTAB is skew-symmetric if A is skew-symmetric
Which of the following is true?
- a)Only S1 is true
- b)Only S2 is true
- c)Both S1 and S2 are true
- d)Both S1 and S2 are false
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Let A and B be the non-singular matrices of same order.Consider the f...
Given that IAI ≠ 0 and IBI ≠ 0
S1:
Consider (BT AB)T = BT AT (BT)T (∵ (AB)T = BTAT)
⇒(BTAB)T = BT AT (BT )T
⇒(BT AB)T = B AT B (∵BT = B)
∴ BT AB is not symmetric ⇔ BT = B
S2:
Let AT = -A
Consider (BTAB)T = BTAT(BT)T
⇒ (BTAB)T = BT(-A)B
⇒(BTAB)T = -BT AB
∴ BT AB is skew-symmetric ⇔ AT = -A
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Let A and B be the non-singular matrices of same order.Consider the f...
Statement S1: BT AB is symmetric if and only if BT = B
To prove this statement, let's assume that BT AB is symmetric. This means that (BT AB)^T = BT AB.
Now, let's calculate the transpose of BT AB:
(BT AB)^T = (AB)^T (BT)^T = B^T A^T B^T
Since BT AB is symmetric, we have:
BT AB = (B^T A^T B^T)^T
Expanding the transpose, we get:
BT AB = (B^T)^T (A^T)^T (B^T)^T = B A B
From the above equation, we can conclude that:
BT AB = B A B
Comparing this equation with the given equation BT AB = BT, we can say that BT = B.
Therefore, the statement S1 is true.
Statement S2: BT AB is skew-symmetric if A is skew-symmetric
To prove this statement, let's assume that A is skew-symmetric. This means that A^T = -A.
Now, let's calculate the transpose of BT AB:
(BT AB)^T = (AB)^T (BT)^T = B^T A^T B^T
Since A is skew-symmetric, we have:
A^T = -A
Expanding the transpose, we get:
(BT AB)^T = B^T (-A) B^T = -B^T A B^T
Comparing this equation with the given equation -(BT AB) = BT, we can see that the two equations do not match.
Therefore, the statement S2 is false.
Hence, the correct answer is option 'B' where only statement S2 is true.
To prove this statement, let's assume that BT AB is symmetric. This means that (BT AB)^T = BT AB.
Now, let's calculate the transpose of BT AB:
(BT AB)^T = (AB)^T (BT)^T = B^T A^T B^T
Since BT AB is symmetric, we have:
BT AB = (B^T A^T B^T)^T
Expanding the transpose, we get:
BT AB = (B^T)^T (A^T)^T (B^T)^T = B A B
From the above equation, we can conclude that:
BT AB = B A B
Comparing this equation with the given equation BT AB = BT, we can say that BT = B.
Therefore, the statement S1 is true.
Statement S2: BT AB is skew-symmetric if A is skew-symmetric
To prove this statement, let's assume that A is skew-symmetric. This means that A^T = -A.
Now, let's calculate the transpose of BT AB:
(BT AB)^T = (AB)^T (BT)^T = B^T A^T B^T
Since A is skew-symmetric, we have:
A^T = -A
Expanding the transpose, we get:
(BT AB)^T = B^T (-A) B^T = -B^T A B^T
Comparing this equation with the given equation -(BT AB) = BT, we can see that the two equations do not match.
Therefore, the statement S2 is false.
Hence, the correct answer is option 'B' where only statement S2 is true.
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Let A and B be the non-singular matrices of same order.Consider the following statements:S1:BT AB is symmetric if and only if BT = BS2: BTAB is skew-symmetric if A is skew-symmetricWhich of the following is true?a)Only S1 is trueb)Only S2 is truec)Both S1 and S2 are trued)Both S1 and S2 are falseCorrect answer is option 'B'. Can you explain this answer?
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Let A and B be the non-singular matrices of same order.Consider the following statements:S1:BT AB is symmetric if and only if BT = BS2: BTAB is skew-symmetric if A is skew-symmetricWhich of the following is true?a)Only S1 is trueb)Only S2 is truec)Both S1 and S2 are trued)Both S1 and S2 are falseCorrect answer is option 'B'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Let A and B be the non-singular matrices of same order.Consider the following statements:S1:BT AB is symmetric if and only if BT = BS2: BTAB is skew-symmetric if A is skew-symmetricWhich of the following is true?a)Only S1 is trueb)Only S2 is truec)Both S1 and S2 are trued)Both S1 and S2 are falseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A and B be the non-singular matrices of same order.Consider the following statements:S1:BT AB is symmetric if and only if BT = BS2: BTAB is skew-symmetric if A is skew-symmetricWhich of the following is true?a)Only S1 is trueb)Only S2 is truec)Both S1 and S2 are trued)Both S1 and S2 are falseCorrect answer is option 'B'. Can you explain this answer?.
Let A and B be the non-singular matrices of same order.Consider the following statements:S1:BT AB is symmetric if and only if BT = BS2: BTAB is skew-symmetric if A is skew-symmetricWhich of the following is true?a)Only S1 is trueb)Only S2 is truec)Both S1 and S2 are trued)Both S1 and S2 are falseCorrect answer is option 'B'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Let A and B be the non-singular matrices of same order.Consider the following statements:S1:BT AB is symmetric if and only if BT = BS2: BTAB is skew-symmetric if A is skew-symmetricWhich of the following is true?a)Only S1 is trueb)Only S2 is truec)Both S1 and S2 are trued)Both S1 and S2 are falseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A and B be the non-singular matrices of same order.Consider the following statements:S1:BT AB is symmetric if and only if BT = BS2: BTAB is skew-symmetric if A is skew-symmetricWhich of the following is true?a)Only S1 is trueb)Only S2 is truec)Both S1 and S2 are trued)Both S1 and S2 are falseCorrect answer is option 'B'. Can you explain this answer?.
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Let A and B be the non-singular matrices of same order.Consider the following statements:S1:BT AB is symmetric if and only if BT = BS2: BTAB is skew-symmetric if A is skew-symmetricWhich of the following is true?a)Only S1 is trueb)Only S2 is truec)Both S1 and S2 are trued)Both S1 and S2 are falseCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for Let A and B be the non-singular matrices of same order.Consider the following statements:S1:BT AB is symmetric if and only if BT = BS2: BTAB is skew-symmetric if A is skew-symmetricWhich of the following is true?a)Only S1 is trueb)Only S2 is truec)Both S1 and S2 are trued)Both S1 and S2 are falseCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of Let A and B be the non-singular matrices of same order.Consider the following statements:S1:BT AB is symmetric if and only if BT = BS2: BTAB is skew-symmetric if A is skew-symmetricWhich of the following is true?a)Only S1 is trueb)Only S2 is truec)Both S1 and S2 are trued)Both S1 and S2 are falseCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
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