Matrix inversion using decomposition Why do I need $LU$, $QR$ and other matrix decomposition tech to find the inverse of a matrix?
Please if possible provide a ref for the answer.
Thanks in advance

If you have the $A=LU$ decomposition, then you could get the the $A^{-1}$ inverse matrix of course with $A^{-1}=U^{-1}L^{-1}$. Computer algebra systems often use the method to calculate the inverse matrix. You can read about it at LU decomposotion wikipedia article. This paper also write about it. You can find details about it also in the following book.

Gene H. Golub – Charles F. Van Loan: Matrix Computations. Third
  Edition. The Johns Hopkins University Press, Baltimore and London,
  1996.
  at page 121.

If you have the $A=QR$ decomposition, you get analogous $A^{-1}=R^{-1}Q^{-1}$. Because $Q$ is orthogonal, you get $A^{-1}=R^{-1}Q^{-1}=R^{-1}Q^{T}$. You can read more about it in the answer.
