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

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*