##### Exercise6

In practice, one rarely finds the inverse of a matrix $$A\text{.}$$ It requires considerable effort to compute, and we can solve any equation of the form $$A\xvec = \bvec$$ using an $$LU$$ factorization, which means that the inverse isn't necessary. In any case, the best way to compute an inverse is using an $$LU$$ factorization, as this exericse demonstrates.

1. Suppose that $$PA = LU\text{.}$$ Explain why $$A^{-1} = U^{-1}L^{-1}P\text{.}$$

Since $$L$$ and $$U$$ are triangular, finding their inverses is relatively efficient. That makes this an effective means of finding $$A^{-1}\text{.}$$

2. Consider the matrix

\begin{equation*} A = \left[\begin{array}{rrr} 3 \amp 4 \amp -1 \\ 2 \amp 4 \amp 1 \\ -3 \amp 1 \amp 4 \\ \end{array}\right] \text{.} \end{equation*}

Find the $$LU$$ factorization of $$A$$ and use it to find $$A^{-1}\text{.}$$

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