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To find the inverse of a matrix using LU decomposition with $u_{11} = u_{22} = u_{33} = 1$, we start by decomposing the given matrix $A$ into a lower triangular matrix $L$ and an upper triangular matrix $U$. Since $U$ is chosen such that its diagonal entries are all 1, we write $A = LU$, where $L = \begin{bmatrix} l_{11} & 0 & 0 \\ l_{21} & l_{22} ____ _______ ____ ________ ____ _________ _________ _________ ____ _________.
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