Activity 5.1.3

We will consider the matrix $$A = \left[\begin{array}{rrr} 1 & 2 & 1 \\ -2 & -3 & -2 \\ 3 & 7 & 4 \\ \end{array}\right]$$ and begin performing Gaussian elimination without using partial pivoting.

1. Perform two row replacement operations to find the row equivalent matrix $$A' = \left[\begin{array}{rrr} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ \end{array}\right]\text{.}$$ Find elementary matrices $E_1$ and $E_2$ that perform these two operations so that $E_2E_1 A = A'$.

2. Perform a third row replacement to find the upper triangular matrix $$U = \left[\begin{array}{rrr} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}\right]\text{.}$$ Find the elementary matrix $E_3$ such that $E_3E_2E_1A = U$.

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

4. Suppose that we want to solve the equation $A\xvec = \bvec = \threevec4{-7}{12}$. We will write $$A\xvec = LU\xvec = L(U\xvec) = \bvec$$ and introduce an unknown vector $\cvec$ such that $U\xvec = \cvec$. Find $\cvec$ by noting that $L\cvec = \bvec$ and solving this equation.

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

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