{
a
1
,
1
x
1
+
a
1
,
2
x
2
+
.
.
.
+
a
1
,
n
x
n
=
b
1
a
2
,
1
x
2
+
a
2
,
2
x
2
+
.
.
.
+
a
2
,
n
x
n
=
b
2
.
.
.
a
m
,
1
x
2
+
a
m
,
2
x
2
+
.
.
.
+
a
m
,
n
x
n
=
b
m
\left\{ \begin{aligned} a_{1,1} x_1 + a_{1,2} x_2 + ... + a_{1,n} x_n &= b_1 \\ a_{2,1} x_2 + a_{2,2} x_2 + ... + a_{2,n} x_n &= b_2 \\ ... \\ a_{m,1} x_2 + a_{m,2} x_2 + ... + a_{m,n} x_n &= b_m \end{aligned} \right.
n
n
m
m
x
1
,
x
2
,
.
.
.
x
n
x_1, x_2, ... x_n
a
i
,
j
(
1
≤
i
≤
n
,
1
≤
j
≤
m
)
a_{i,j} \; \left( {1 \leq i \leq n}, {1 \leq j \leq m} \right)
b
i
(
1
≤
i
≤
n
)
b_{i} \; \left( {1 \leq i \leq n} \right)
(
L
i
↔
L
j
L_i \leftrightarrow L_j
)
(
L
i
←
λ
L
i
L_i \leftarrow \lambda L_i
)
(
L
i
←
L
i
+
λ
L
j
L_i \leftarrow L_i + \lambda L_j
)
{
1
x
+
y
+
z
=
1
2
x
+
3
y
+
4
z
=
5
9
x
+
8
y
+
7
z
=
6
⟶
L
2
←
L
2
-
2
L
1
,
L
3
←
L
3
-
9
L
1
{
x
+
y
+
z
=
1
y
+
2
z
=
3
-
y
-
2
z
=
-
3
\left\{ \begin{aligned} {\color{red} 1}x + y + z &= 1 \\ 2x + 3y + 4z &= 5 \\ 9x + 8y + 7z &= 6 \\ \end{aligned} \right. \; \underset{ {L_2 \leftarrow L_2 - 2 L_1}, {L_3 \leftarrow L_3 - 9 L_1}}{\longrightarrow} \; \left\{ \begin{aligned} x + y + z &= 1 \\ y + 2z &= 3 \\ -y - 2z &= -3 \\ \end{aligned} \right.