##### Activity5.1.3

We will consider the matrix

\begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 2 \amp 1 \\ -2 \amp -3 \amp -2 \\ 3 \amp 7 \amp 4 \\ \end{array}\right] \end{equation*}

and begin performing Gaussian elimination.

1. Perform two row replacement operations to find the row equivalent matrix

\begin{equation*} A' = \left[\begin{array}{rrr} 1 \amp 2 \amp 1 \\ 0 \amp 1 \amp 0 \\ 0 \amp 1 \amp 1 \\ \end{array}\right]\text{.} \end{equation*}

Find elementary matrices $$E_1$$ and $$E_2$$ that perform these two operations so that $$E_2E_1 A = A'\text{.}$$

2. Perform a third row replacement to find the upper triangular matrix

\begin{equation*} U = \left[\begin{array}{rrr} 1 \amp 2 \amp 1 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end{array}\right]\text{.} \end{equation*}

Find the elementary matrix $$E_3$$ such that $$E_3E_2E_1A = U\text{.}$$

3. We can write $$A=E_1^{-1}E_2^{-1}E_3^{-1} U\text{.}$$ Find the inverse matrices $$E_1^{-1}\text{,}$$ $$E_2^{-1}\text{,}$$ and $$E_3^{-1}$$ and the product $$L = E_1^{-1}E_2^{-1}E_3^{-1}\text{.}$$ Then verify that $$A=LU\text{.}$$

4. Suppose that we want to solve the equation $$A\xvec = \bvec = \threevec4{-7}{12}\text{.}$$ We will write

\begin{equation*} A\xvec = LU\xvec = L(U\xvec) = \bvec \end{equation*}

and introduce an unknown vector $$\cvec$$ such that $$U\xvec = \cvec\text{.}$$ Find $$\cvec$$ by noting that $$L\cvec = \bvec$$ and solving this equation.

5. Now that we have found $$\cvec\text{,}$$ find $$\xvec$$ by solving $$U\xvec = \cvec\text{.}$$

6. Use the factorization $$A=LU$$ and this two-step process, solve the equation $$A\xvec = \threevec{2}{-2}{7}\text{.}$$

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