# Is this correct about matrices

#### Yegor

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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#### matt grime

Homework Helper
who says it is incorrect?

#### Hurkyl

Staff Emeritus
Gold Member
$$(BA)^{-1}=A^{-1}B^{-1}$$
Yes, provided A and B are both invertible matrices...

#### Yegor

$$(A^{T}A)^{-1}A^{T}$$ 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 $$x=A^{-1}b$$

#### Yegor

Oh, yes. Now i see. Thank you very much, Hurky!