# Find the LU-Factorization of the matrix A below A=[(2,1,-1),(-2,0,3),(2,1,-4),(4,1,-4),(6,5,-2)]

Question
Polynomial factorization
Find the LU-Factorization of the matrix A below
$$\displaystyle{A}={\left[\begin{array}{ccc} {2}&{1}&-{1}\\-{2}&{0}&{3}\\{2}&{1}&-{4}\\{4}&{1}&-{4}\\{6}&{5}&-{2}\end{array}\right]}$$

2021-02-25
Step 1
Given
$$\displaystyle{\left[\begin{array}{ccc} {2}&{1}&-{1}\\-{2}&{0}&{3}\\{2}&{1}&-{4}\\{4}&{1}&-{4}\\{6}&{5}&-{2}\end{array}\right]}$$
Step 2
Solution
L=Lower triangular matrix
U=Upper triangular matrix
$$\displaystyle{A}={\left[\begin{array}{ccc} {2}&{1}&-{1}\\-{2}&{0}&{3}\\{2}&{1}&-{4}\\{4}&{1}&-{4}\\{6}&{5}&-{2}\end{array}\right]}\sim{\left[\begin{array}{ccc} {2}&{1}&-{1}\\{0}&{1}&{2}\\{0}&{0}&-{3}\\{0}&-{1}&-{2}\\{0}&{2}&{1}\end{array}\right]}{\left(\begin{array}{c} {R}_{{2}}\rightarrow{R}_{{2}}+{R}_{{1}}\\{R}_{{3}}\rightarrow{R}_{{3}}-{R}_{{1}}\\{R}_{{4}}\rightarrow{R}_{{4}}-{2}{R}_{{1}}\\{R}_{{5}}\rightarrow{R}_{{5}}-{3}{R}_{{1}}\end{array}\right)}$$
$$\displaystyle\sim{\left[\begin{array}{ccc} {2}&{1}&-{1}\\{0}&{1}&{2}\\{0}&{0}&-{3}\\{0}&{0}&{0}\\{0}&{0}&-{3}\end{array}\right]}{R}_{{4}}\rightarrow{R}_{{4}}+{R}_{{2}},{R}_{{5}}\rightarrow{R}_{{5}}-{2}{R}_{{2}}$$
$$\displaystyle\sim{\left[\begin{array}{ccc} {2}&{1}&-{1}\\{0}&{1}&{2}\\{0}&{0}&-{3}\\{0}&{0}&{0}\\{0}&{0}&{0}\end{array}\right]}{R}_{{5}}\rightarrow{R}_{{5}}-{R}_{{3}}$$
$$\displaystyle{U}={\left[\begin{array}{ccc} {2}&{1}&-{1}\\{0}&{1}&{2}\\{0}&{0}&-{3}\\{0}&{0}&{0}\\{0}&{0}&{0}\end{array}\right]}$$
Step 3
And lower triangular matrix that is L is calculated by the entries are the number in the 0 respectively.
i.e., $$\displaystyle{L}={\left[\begin{array}{ccc} {1}&{0}&{0}\\-{1}&{1}&{0}\\{1}&{0}&{1}\\{2}&-{1}&{0}\\{3}&{2}&{1}\end{array}\right]}$$

### Relevant Questions

find an LU factorization of the given matrix.
$$\displaystyle{\left[\begin{array}{ccc} {2}&{2}&-{1}\\{4}&{0}&{4}\\{3}&{4}&{4}\end{array}\right]}$$
Find an LU factorization of the matrix A (with L unit lower triangular).
$$\displaystyle{A}={\left[\begin{array}{cc} {5}&{4}\\-{4}&-{3}\end{array}\right]}$$
L-?
U-?
Find an LU factorization of the matrix A (with L unit lower triangular).
$$\displaystyle{A}={\left[\begin{array}{ccc} -{4}&{0}&{4}\\{12}&{2}&-{9}\\{12}&{8}&{9}\end{array}\right]}$$
L-?
U-?
Find an LU factorization of $$\displaystyle{A}={\left[\begin{array}{cccc} {h}&-{4}&-{2}&{10}\\{h}&-{9}&{4}&{2}\\{0}&{0}&-{4}&{2}\\{0}&{1}&{4}&{4}\\{0}&{0}&{0}&\frac{{h}}{{2}}\end{array}\right]}$$.
h=102
Use the prime factorizations $$\displaystyle{a}={2}^{{4}}\times{3}^{{4}}\times{5}^{{2}}\times{7}^{{3}}{\quad\text{and}\quad}{b}={2}^{{2}}\times{3}\times{5}^{{3}}\times{11}$$ to find the prime factorization of the following.
(a) LCM(a, b)
(b) GCF(a, b)
Use the factorization theorem to determine whether $$\displaystyle{x}−\frac{{1}}{{2}}$$ is a factor
of $$\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{4}}−{x}^{{3}}+{2}{x}−{1}$$.
$$\displaystyle{\left(\frac{{-{3}}}{{{2}\sqrt{{5}}}}\right)},{\left(-\frac{{1}}{{2}}\right)}$$
$$P(x) = x^{4}+2x^{2}+1$$
Need to calculate:The factorization of $$3p^{3}-2p^{2}-9p+6$$.
Need to calculate:The factorization of $$x^{3}+3x^2+2x+6$$