\begin{aligned} x_{1} + 2x_{2} + x_{3} &= -1 \\ -3x_{1} + x_{2} + x_{3} &= 0 \\ x_{1} + 2x_{3} &= 1 \\ \end{aligned} \implies \overset{A}{ \begin{bmatrix} 1 & 2 & 1 \\ -3 & 1 & 1 \\ 1 & 0 & 3 \\ \end{bmatrix} } \overset{x}{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} } = \overset{b}{ \begin{bmatrix} -1 \\ 0 \\ 1 \\ \end{bmatrix} }

Using the formula Row_{i} := Row_{i} - (\frac{A_{ik}}{A_{kk}}) \cdot Row_{k}:

A_{ik} = A_{21} hence i = 2 and k = 1

Row_{2} := Row_{2} - (\frac{A_{21}}{A_{11}}) \cdot Row_{1}

= \begin{bmatrix}-3 & 1 & 1\end{bmatrix} - (\frac{-3}{1}) \cdot \begin{bmatrix}1 & 2 & 1\end{bmatrix}

= \begin{bmatrix}0 & 7 & 4\end{bmatrix}


\begin{bmatrix} 1 & 2 & 1 \\ 0 & 7 & 4 \\ 1 & 0 & 3 \\ \end{bmatrix}

A_{ik} = A_{31} hence i = 3 and k = 1

Row_{3} := Row_{3} - (\frac{A_{31}}{A_{11}}) \cdot Row_{1}

= \begin{bmatrix}1 & 0 & 3\end{bmatrix} - (\frac{1}{1}) \cdot \begin{bmatrix}1 & 2 & 1\end{bmatrix}

= \begin{bmatrix}0 & -2 & 2\end{bmatrix}


\begin{bmatrix} 1 & 2 & 1 \\ 0 & 7 & 4 \\ 0 & -2 & 2 \\ \end{bmatrix}

A_{ik} = A_{32} hence i = 3 and k = 2

Row_{3} := Row_{3} - (\frac{A_{32}}{A_{22}}) \cdot Row_{2}

= \begin{bmatrix}0 & -2 & 2\end{bmatrix} - (\frac{-2}{7}) \cdot \begin{bmatrix}0 & 7 & 4\end{bmatrix}

= \begin{bmatrix}0 & 0 & \frac{22}{7}\end{bmatrix}


\begin{bmatrix} 1 & 2 & 1 \\ 0 & 7 & 4 \\ 0 & 0 & \frac{22}{7} \\ \end{bmatrix}


\overset{A}{ \begin{bmatrix} 1 & 2 & 1 \\ -3 & 1 & 1 \\ 1 & 0 & 3 \\ \end{bmatrix} } = \overset{L}{ \begin{bmatrix} 1 & 0 & 0 \\ -3 & 1 & 0 \\ 1 & \frac{-2}{7} & 1 \\ \end{bmatrix} } \overset{U}{ \begin{bmatrix} 1 & 2 & 1 \\ 0 & 7 & 4\\ 0 & 0 & \frac{22}{7} \\ \end{bmatrix} }


Lw = b \implies \overset{L}{ \begin{bmatrix} 1 & 0 & 0 \\ -3 & 1 & 0 \\ 1 & \frac{-2}{7} & 1 \\ \end{bmatrix} } \overset{w}{ \begin{bmatrix} w_{1} \\ w_{2} \\ w_{3} \\ \end{bmatrix} } = \overset{b}{ \begin{bmatrix} -1 \\ 0 \\ 1 \\ \end{bmatrix} } 

for j = 1 to n:
    j = 1
    w1 = b1 / L11 = -1 / 1 = -1
    for i = j + 1 to n:
        i = 2
        b2 = b2 - L21 * w1 = 0 - (-3) * (-1) = -3
        i = 3
        b3 = b3 - L31 * w1 = 1 - 1 * (-1) = 2
    j = 2
    w2 = b2 / L22 = -3 / 1 = -3
    for i = j + 1 to n:
        i = 3
        b3 = b3 - L32 * w2 = 2 - (-2 / 7) * (-3) = 8 / 7
    j = 3
    w3 = b3 / L33 = (8 / 7) / 1 = 8 / 7
    for i = j + 1 to n:
        i = 4 -> skip


Ux = w \implies \overset{U}{ \begin{bmatrix} 1 & 2 & 1 \\ 0 & 7 & 4 \\ 0 & 0 & \frac{22}{7} \\ \end{bmatrix} } \overset{x}{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} } = \overset{w}{ \begin{bmatrix} w_{1} \\ w_{2} \\ w_{3} \\ \end{bmatrix} } = \overset{w}{ \begin{bmatrix} -1 \\ -3 \\ \frac{8}{7} \\ \end{bmatrix} }

Paul Laffitte 6222808130