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
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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.