Exercise 6

In practice, one rarely finds the inverse of a matrix $A$. 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 exercise demonstrates.

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

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

2. Consider the matrix $$A = \left[\begin{array}{rrr} 3 & 4 & -1 \\ 2 & 4 & 1 \\ -3 & 1 & 4 \\ \end{array}\right]$$. Find the $LU$ factorization of $A$ and use it to find $A^{-1}$.