# Compute the LU factorization of each of the following matrices. begin{bmatrix}2 & 4 -2 & 1 end{bmatrix}

Compute the LU factorization of each of the following matrices.
$\left[\begin{array}{cc}2& 4\\ -2& 1\end{array}\right]$
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Step 1 : Given:
A matrix $\left[\begin{array}{cc}2& 4\\ -2& 1\end{array}\right]$ We have to compute the LU factorization of this matrix.
Step 2 : Formula Used
LU Factorization: If we have a matrix A, then an upper triangular matrix U obtained without pivoting under Gaussian elimination method, and there exists lower triangular matrix L s.t. A = LU.
Step 3: Calculation
$⇒\left[\begin{array}{cc}2& 4\\ -2& 1\end{array}\right]$
Use Gaussian Eliminatin Method , ${R}_{2}←{R}_{2}-\left(-1\right){R}_{1}$
$⇒\left[\begin{array}{cc}2& 4\\ 0& 5\end{array}\right]$
$\therefore U=\left[\begin{array}{cc}2& 4\\ 0& 5\end{array}\right]$
Now , use gaussian elimination with 1st column , we get
$⇒\left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right]$
$\therefore L=\left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right]$
Hence,
$\left[\begin{array}{cc}2& 4\\ -2& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right]\left[\begin{array}{cc}2& 4\\ 0& 5\end{array}\right]=LU$
This is the LU factorization of the given matrix.