I have been studying multiple regression recently and $\hat{\beta}$ is derived as $(\mathbf{X}^\top \mathbf{X})^{-1}\mathbf{X}^\top$ where $\mathbf{X}$ is an any matrix. What is wrong if I simplify $\hat{\beta}$ as $(\mathbf{X}^\top \mathbf{X})^{-1}\mathbf{X}^\top$ = $\mathbf{X}^{-1}(\mathbf{X}^\top)^{-1}\mathbf{X}^\top$ = $\mathbf{X}^{-1}$? Thank you.
2026-02-22 23:40:25.1771803625
Decomposition of the product of matrices
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$X$ and $X^T$ need not be square matrices and hence their inverse are not defined.
However, you are right that if $X$ is invertible and you are solving the system
$$X\hat{\beta} = y,$$
then we have
$$\hat{\beta} = X^{-1}y.$$