Find the LU decomposition of the following matrices.
1). \( \left[\begin{array}{l l l}1&1&1\\ 3&-2&2\\ 1&0&0\end{array}\right].\)
2). \(\left[\begin{array}{lll}0 & 3 & 1 \\ 1 & 1 & 2 \\ 0 & 0 & 5\end{array}\right]\)
3). \(\left[\begin{array}{ccc}1 & 1 & 1 \\ 4 & 3 & -1 \\ 3 & 5 & 3\end{array}\right]\)
4). \(\left[{\begin{array}{r r r}{2}&{3}&{4}\\ {7}&{0}&{0}\\ {1}&{1}&{2}\end{array}}\right]\)
5). \({\left[\begin{array}{l l l}{1}&{3}&{1}\\ {1}&{4}&{2}\\ {-1}&{-2}&{3}\end{array}\right]}.\)
Answers:
1). \(L=\left[\begin{array}{ccc}1 & 0 & 0 \\ 3 & 1 & 0 \\ 1 & \frac{1}{5} & 1\end{array}\right]\)
\(U=\left[\begin{array}{ccc}1 & 1 & 1 \\ 0 & -5 & -1 \\ 0 & 0 & -\frac{4}{5}\end{array}\right]\)
2). \(L=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\)
\(U=\left[\begin{array}{lll}1 & 1 & 2 \\ 0 & 3 & 1 \\ 0 & 0 & 5\end{array}\right]\)
3). \(L=\left[\begin{array}{ccc}1 & 0 & 0 \\ 4 & 1 & 0 \\ 3 & -2 & 1\end{array}\right]\) \(U=\left[\begin{array}{ccc}1 & 1 & 1 \\ 0 & -1 & -5 \\ 0 & 0 & -10\end{array}\right]\)
4). \(L=\left[\begin{array}{ccc}1 & 0 & 0 \\ \frac{7}{2} & 1 & 0 \\ \frac{1}{2} & \frac{1}{21} & 1\end{array}\right]\)
\(U=\left[\begin{array}{ccc}2 & 3 & 4 \\ 0 & -\frac{21}{2} & -14 \\ 0 & 0 & \frac{2}{3}\end{array}\right]\)
5). \(L=\left[\begin{array}{ccc}1 & 0 & 0 \\ 1 & 1 & 0 \\ -1 & 1 & 1\end{array}\right]\)
\(U=\left[\begin{array}{lll}1 & 3 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 3\end{array}\right]\)
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