# Find an LU factorization of A = [(h,-4,-2,10),(h,-9,4,2),(0,0,-4,2),(0,1,4,4),(0,0,0,h/2)]. h=102 Question
Polynomial factorization 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 2021-02-14
Step 1
$$\displaystyle{A}={\left[\begin{array}{cccc} {102}&-{4}&-{2}&{10}\\{102}&-{9}&{4}&{2}\\{0}&{0}&-{4}&{2}\\{0}&{1}&{4}&{4}\\{0}&{0}&{0}&{51}\end{array}\right]}$$
In this problem, we have to find matrices L(lower triangular) and U(upper triangular) for which A=LU
Initial Matrix: $$\displaystyle{\left[\begin{array}{ccccc} {1}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}&{1}\end{array}\right]}{\left[\begin{array}{cccc} {102}&-{4}&-{2}&{10}\\{102}&-{9}&{4}&{2}\\{0}&{0}&-{4}&{2}\\{0}&{1}&{4}&{4}\\{0}&{0}&{0}&{51}\end{array}\right]}$$
Using some row transformations we convert this initial matrix into LU factorization,
where L will the matrix having all diagonal entries 1, and entries above diagonal 0.
Step 2
by some row transformations, we get the matrix L and matrix U
$$\displaystyle{L}={\left[\begin{array}{ccccc} {1}&{0}&{0}&{0}&{0}\\{1}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&-\frac{{1}}{{5}}&-\frac{{13}}{{10}}&{1}&{0}\\{0}&{0}&{0}&\frac{{51}}{{5}}&{1}\end{array}\right]}$$
$$\displaystyle{U}={\left[\begin{array}{cccc} {102}&-{4}&-{2}&{10}\\{0}&-{5}&{6}&-{8}\\{0}&{0}&-{4}&{2}\\{0}&{0}&{0}&{5}\\{0}&{0}&{0}&{0}\end{array}\right]}$$
If we multiply L and U, we will get the same matrix A.

### Relevant Questions 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 the given matrix.
$$\displaystyle{\left[\begin{array}{ccc} {2}&{2}&-{1}\\{4}&{0}&{4}\\{3}&{4}&{4}\end{array}\right]}$$ Find the LU-Factorization of the matrix A below
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$$\displaystyle{A}={\left[\begin{array}{cc} {5}&{4}\\-{4}&-{3}\end{array}\right]}$$
L-?
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$$P(x) = x^{4}+2x^{2}+1$$ 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. Use the factorization theorem to determine whether $$\displaystyle{x}−\frac{{1}}{{2}}$$ is a factor
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