Why Is (A^{T}A)^{-1}A^{T}=A^{-1}(A^{T})^{-1}A^{T}=A^{-1} Incorrect?

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The discussion clarifies that the expression (A^{T}A)^{-1}A^{T} is not equal to A^{-1} due to the context of least squares approximation in inconsistent linear systems. While (AB)C = A(BC) and (BA)^{-1} = A^{-1}B^{-1} are valid for invertible matrices, the specific case of (A^{T}A)^{-1}A^{T} arises in scenarios where A is not necessarily invertible. The correct interpretation is that (A^{T}A)^{-1}A^{T} provides the best approximation for solutions to Ax = b, rather than directly inverting A. This distinction is crucial for understanding matrix operations in linear algebra.
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I know that for matrices A, B and C is correct to write: (AB)C=A(BC)
Also (BA)^{-1}=A^{-1}B^{-1}
Why (A^{T}A)^{-1}A^{T}=A^{-1}(A^{T})^{-1}A^{T}=A^{-1} is not correct?
 
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who says it is incorrect?
 
(BA)^{-1}=A^{-1}B^{-1}

Yes, provided A and B are both invertible matrices...
 
(A^{T}A)^{-1}A^{T} such expression comes in chapter about least squares aproximation.
e.g. if we have inconsistent linear system Ax=b, then x=(A^{T}A)^{-1}A^{T}b is best approximation. It is not equal to x=A^{-1}b
 
Oh, yes. Now i see. Thank you very much, Hurky!
 

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