# Find an LU factorization of the matrix A (with L unit lower triangular). A=[(-4,0,4),(12,2,-9),(12,8,9)] L-? U-?

Question
Polynomial factorization
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-?

2020-12-13
Step 1
The given matrix is $$\displaystyle{A}={\left[\begin{array}{ccc} -{4}&{0}&{4}\\{12}&{2}&-{9}\\{12}&{8}&{9}\end{array}\right]}$$.
Using Gaussian Elimination method,
$$\displaystyle{A}={\left[\begin{array}{ccc} -{4}&{0}&{4}\\{0}&{2}&{3}\\{12}&{8}&{9}\end{array}\right]}{R}_{{2}}\rightarrow{R}_{{2}}+{3}{R}_{{1}}$$
$$\displaystyle={\left[\begin{array}{ccc} -{4}&{0}&{4}\\{0}&{2}&{3}\\{0}&{8}&{21}\end{array}\right]}{R}_{{3}}\rightarrow{R}_{{3}}+{3}{R}_{{1}}$$
$$\displaystyle={\left[\begin{array}{ccc} -{4}&{0}&{4}\\{0}&{2}&{3}\\{0}&{0}&{9}\end{array}\right]}{R}_{{3}}\rightarrow{R}_{{3}}-{4}{R}_{{2}}$$
Step 2
Since, $$\displaystyle{U}={\left[\begin{array}{ccc} -{4}&{0}&{4}\\{0}&{2}&{3}\\{0}&{0}&{9}\end{array}\right]}$$.
L is just up of the multipliers we used in Gaussian elimination with ls on the diagonal.
Since, $$\displaystyle{L}={\left[\begin{array}{ccc} {1}&{0}&{0}\\-{3}&{1}&{0}\\-{3}&{4}&{1}\end{array}\right]}$$.
Hence Lu decomposition for A is
$$\displaystyle{A}={\left[\begin{array}{ccc} -{4}&{0}&{4}\\{12}&{2}&-{9}\\{12}&{8}&{9}\end{array}\right]}={\left[\begin{array}{ccc} {1}&{0}&{0}\\-{3}&{1}&{0}\\-{3}&{4}&{1}\end{array}\right]}{\left[\begin{array}{ccc} -{4}&{0}&{4}\\{0}&{2}&{3}\\{0}&{0}&{9}\end{array}\right]}={L}{U}$$.

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