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-?

Find an LU factorization of the matrix A (with L unit lower triangular).
$A=\left[\begin{array}{ccc}-4& 0& 4\\ 12& 2& -9\\ 12& 8& 9\end{array}\right]$
L-?
U-?
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Step 1
The given matrix is $A=\left[\begin{array}{ccc}-4& 0& 4\\ 12& 2& -9\\ 12& 8& 9\end{array}\right]$.
Using Gaussian Elimination method,
$A=\left[\begin{array}{ccc}-4& 0& 4\\ 0& 2& 3\\ 12& 8& 9\end{array}\right]{R}_{2}\to {R}_{2}+3{R}_{1}$
$=\left[\begin{array}{ccc}-4& 0& 4\\ 0& 2& 3\\ 0& 8& 21\end{array}\right]{R}_{3}\to {R}_{3}+3{R}_{1}$
$=\left[\begin{array}{ccc}-4& 0& 4\\ 0& 2& 3\\ 0& 0& 9\end{array}\right]{R}_{3}\to {R}_{3}-4{R}_{2}$
Step 2
Since, $U=\left[\begin{array}{ccc}-4& 0& 4\\ 0& 2& 3\\ 0& 0& 9\end{array}\right]$.
L is just up of the multipliers we used in Gaussian elimination with ls on the diagonal.
Since, $L=\left[\begin{array}{ccc}1& 0& 0\\ -3& 1& 0\\ -3& 4& 1\end{array}\right]$.
Hence Lu decomposition for A is
$A=\left[\begin{array}{ccc}-4& 0& 4\\ 12& 2& -9\\ 12& 8& 9\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ -3& 1& 0\\ -3& 4& 1\end{array}\right]\left[\begin{array}{ccc}-4& 0& 4\\ 0& 2& 3\\ 0& 0& 9\end{array}\right]=LU$.