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If A be any matrix then AAT and ATA are
- a)Skew Symmetric
- b)Both are Symmetric
- c)Hermitian
- d)Skew Hermitian
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If A be any matrix then AAT and ATA area)Skew Symmetricb)Both are Symm...
Let A be real matrix , then take
(AAT)T = (AT)AT = AAT
⇒ AAT is Symmetric
now
(ATA)T = AT(AT)T = ATA
⇒ ATA is Symmetric
Most Upvoted Answer
If A be any matrix then AAT and ATA area)Skew Symmetricb)Both are Symm...
Explanation:
To determine the properties of the matrices AAT and ATA, we need to understand the definitions of these matrices and the properties they possess.
Definition of AAT:
The matrix AAT is obtained by multiplying matrix A with its transpose. The resulting matrix has the same number of rows as A and the same number of columns as the transpose of A.
Definition of ATA:
The matrix ATA is obtained by multiplying the transpose of matrix A with A. The resulting matrix has the same number of rows as the transpose of A and the same number of columns as A.
Properties of AAT:
When we multiply A by its transpose, the resulting matrix AAT will always be a symmetric matrix. This is because the entry (i, j) in AAT is the dot product of the ith row of A with the jth row of A, which is the same as the dot product of the jth row of A with the ith row of A. Therefore, AAT is symmetric.
Properties of ATA:
Similarly, when we multiply the transpose of A with A, the resulting matrix ATA will also be a symmetric matrix. This is because the entry (i, j) in ATA is the dot product of the ith column of A with the jth column of A, which is the same as the dot product of the jth column of A with the ith column of A. Therefore, ATA is symmetric.
Conclusion:
From the properties stated above, we can conclude that both AAT and ATA are symmetric matrices. Therefore, the correct answer is option 'B' - Both are symmetric.
To determine the properties of the matrices AAT and ATA, we need to understand the definitions of these matrices and the properties they possess.
Definition of AAT:
The matrix AAT is obtained by multiplying matrix A with its transpose. The resulting matrix has the same number of rows as A and the same number of columns as the transpose of A.
Definition of ATA:
The matrix ATA is obtained by multiplying the transpose of matrix A with A. The resulting matrix has the same number of rows as the transpose of A and the same number of columns as A.
Properties of AAT:
When we multiply A by its transpose, the resulting matrix AAT will always be a symmetric matrix. This is because the entry (i, j) in AAT is the dot product of the ith row of A with the jth row of A, which is the same as the dot product of the jth row of A with the ith row of A. Therefore, AAT is symmetric.
Properties of ATA:
Similarly, when we multiply the transpose of A with A, the resulting matrix ATA will also be a symmetric matrix. This is because the entry (i, j) in ATA is the dot product of the ith column of A with the jth column of A, which is the same as the dot product of the jth column of A with the ith column of A. Therefore, ATA is symmetric.
Conclusion:
From the properties stated above, we can conclude that both AAT and ATA are symmetric matrices. Therefore, the correct answer is option 'B' - Both are symmetric.
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If A be any matrix then AAT and ATA area)Skew Symmetricb)Both are Symmetricc)Hermitiand)Skew HermitianCorrect answer is option 'B'. Can you explain this answer?
Question Description
If A be any matrix then AAT and ATA area)Skew Symmetricb)Both are Symmetricc)Hermitiand)Skew HermitianCorrect answer is option 'B'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about If A be any matrix then AAT and ATA area)Skew Symmetricb)Both are Symmetricc)Hermitiand)Skew HermitianCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If A be any matrix then AAT and ATA area)Skew Symmetricb)Both are Symmetricc)Hermitiand)Skew HermitianCorrect answer is option 'B'. Can you explain this answer?.
If A be any matrix then AAT and ATA area)Skew Symmetricb)Both are Symmetricc)Hermitiand)Skew HermitianCorrect answer is option 'B'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about If A be any matrix then AAT and ATA area)Skew Symmetricb)Both are Symmetricc)Hermitiand)Skew HermitianCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If A be any matrix then AAT and ATA area)Skew Symmetricb)Both are Symmetricc)Hermitiand)Skew HermitianCorrect answer is option 'B'. Can you explain this answer?.
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