Matrices and Vectors
Matrix
(
1
3
3
2
1
3
9
-6
4
3
2
6
-4
2
2
)
column matrices
(
3
9
6
)
(
3
-6
-4
)
(
2
4
2
)
(
1
3
2
)
row matrices
(
3
3
2
1
)
(
9
-6
4
3
)
(
6
-4
2
2
)
(
5
-8
1
3
-5
1
-4
7
-1
)
(
2
1
3
1
1
2
-1
3
1
)
(
5
0
0
3
-5
0
-4
7
-1
)
(
0
1
3
0
1
2
0
0
1
)
(
5
0
0
0
-5
0
0
0
-1
)
(
-3
0
0
0
-3
0
0
0
-3
)
(
1
0
0
0
1
0
0
0
1
)
Representing SLE
Standard form
x
1
+
3
x
2
+
3
x
3
+
2
x
4
+
x
5
=
7
3
x
1
+
9
x
2
-
6
x
3
+
4
x
4
+
3
x
5
=
-7
2
x
1
+
6
x
2
-
4
x
3
+
2
x
4
+
2
x
5
=
-4
Augmented Matrix
(
1
3
3
2
1
7
3
9
-6
4
3
-7
2
6
-4
2
2
-4
)
Vector representation
(
1
3
2
)
x
1
+
(
3
9
6
)
x
2
+
(
3
-6
-4
)
x
3
+
(
2
4
2
)
x
4
+
(
1
3
2
)
x
5
=
(
7
-7
-4
)
Matrix Representation
(
1
3
3
2
1
3
9
-6
4
3
2
6
-4
2
2
)
(
x
1
x
2
x
3
x
4
x
5
)
=
(
7
-7
-4
)
Matrix Representation
(
1
3
3
2
1
3
9
-6
4
3
2
6
-4
2
2
)
x
→
=
(
7
-7
-4
)
Gauss Method
Example 01
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
+
x
5
=
7
3
x
1
+
9
x
2
-
6
x
3
+
4
x
4
+
3
x
5
=
-7
2
x
1
+
6
x
2
-
4
x
3
+
2
x
4
+
2
x
5
=
-4
→
ρ
2
′
=
-3
ρ
1
+
ρ
2
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
+
x
5
=
7
-15
x
3
-
2
x
4
=
-28
2
x
1
+
6
x
2
-
4
x
3
+
2
x
4
+
2
x
5
=
-4
(
1
0
0
-3
1
0
0
0
1
)
(
1
3
3
2
1
7
3
9
-6
4
3
-7
2
6
-4
2
2
-4
)
=
(
1
3
3
2
1
9
0
0
-15
-2
0
-28
2
6
-4
2
2
-4
)
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
+
x
5
=
7
-15
x
3
-
2
x
4
=
-28
2
x
1
+
6
x
2
-
4
x
3
+
2
x
4
+
2
x
5
=
-4
→
ρ
3
′
=
-2
ρ
1
+
ρ
3
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
+
x
5
=
7
-15
x
3
-
2
x
4
=
-28
-10
x
3
-2
x
4
=
-18
(
1
0
0
0
1
0
-2
0
1
)
(
1
3
3
2
1
7
0
0
-15
-2
0
-28
2
6
-4
2
2
-4
)
=
(
1
3
3
2
1
7
0
0
-15
-2
0
-28
0
0
-10
-2
0
-18
)
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
+
x
5
=
7
-15
x
3
-
2
x
4
=
-28
-10
x
3
-2
x
4
=
-18
→
ρ
3
′
=
-2
3
ρ
2
+
ρ
3
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
+
x
5
=
7
-15
x
3
-
2
x
4
=
-28
-2
3
x
4
=
2
3
(
1
0
0
0
1
0
0
-2
3
1
)
(
1
3
3
2
1
7
0
0
-15
-2
0
-28
0
0
-10
-2
0
-18
)
=
(
1
3
3
2
1
7
0
0
-15
-2
0
-28
0
0
0
-2
3
0
2
3
)
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
+
x
5
=
7
-15
x
3
-
2
x
4
=
-28
-2
3
x
4
=
2
3
→
ρ
3
′
=
-3
2
ρ
3
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
+
x
5
=
7
-15
x
3
-
2
x
4
=
-28
x
4
=
-1
(
1
0
0
0
1
0
0
0
-3
2
)
(
1
3
3
2
1
7
0
0
-15
-2
0
-28
0
0
0
-2
3
0
2
3
)
=
(
1
3
3
2
1
7
0
0
-15
-2
0
-28
0
0
0
1
0
-1
)
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
+
x
5
=
7
-15
x
3
-
2
x
4
=
-28
x
4
=
-1
→
ρ
1
′
=
ρ
1
-2
ρ
3
{
x
1
+
3
x
2
+
3
x
3
+
x
5
=
9
-15
x
3
-
2
x
4
=
-28
x
4
=
-1
(
1
0
-2
0
1
0
0
0
1
)
(
1
3
3
2
1
7
0
0
-15
-2
0
-28
0
0
0
1
0
-1
)
=
(
1
3
3
0
1
9
0
0
-15
-2
0
-28
0
0
0
1
0
-1
)
{
x
1
+
3
x
2
+
3
x
3
+
x
5
=
9
-15
x
3
-
2
x
4
=
-28
x
4
=
-1
→
ρ
2
′
=
ρ
2
+2
ρ
3
{
x
1
+
3
x
2
+
3
x
3
+
x
5
=
9
-15
x
3
=
-30
x
4
=
-1
(
1
0
0
0
1
2
0
0
1
)
(
1
3
3
0
1
9
0
0
-15
-2
0
-28
0
0
0
1
0
-1
)
=
(
1
3
3
0
1
9
0
0
-15
0
0
-30
0
0
0
1
0
-1
)
{
x
1
+
3
x
2
+
3
x
3
+
x
5
=
9
-15
x
3
=
-30
x
4
=
-1
→
ρ
2
′
=
-1
15
ρ
2
{
x
1
+
3
x
2
+
3
x
3
+
x
5
=
9
x
3
=
2
x
4
=
-1
(
1
0
0
0
-1
15
0
0
0
1
)
(
1
3
3
0
1
9
0
0
-15
0
0
-30
0
0
0
1
0
-1
)
=
(
1
3
3
0
1
9
0
0
1
0
0
2
0
0
0
1
0
-1
)
{
x
1
+
3
x
2
+
3
x
3
+
x
5
=
9
x
3
=
2
x
4
=
-1
→
ρ
1
′
=
ρ
1
-3
ρ
2
{
x
1
+
3
x
2
+
x
5
=
3
x
3
=
2
x
4
=
-1
(
1
-3
0
0
1
0
0
0
1
)
(
1
3
3
0
1
9
0
0
1
0
0
2
0
0
0
1
0
-1
)
=
(
1
3
0
0
1
3
0
0
1
0
0
2
0
0
0
1
0
-1
)
x
1
=
3
-
3
s
1
-
3
s
2
x
2
=
s
1
x
3
=
2
x
4
=
-1
x
5
=
s
2
or
{
(
3
0
2
-1
0
)
⏟
particular
+
(
-3
1
0
0
0
)
s
1
+
(
-1
0
0
0
1
)
s
2
⏟
homogeneous
∣
s
1
,
s
2
∈
ℝ
}
Gauss Method
Example 02
{
x
3
+
2
x
4
=
3
x
1
+
3
x
2
+
3
x
3
+
2
x
4
=
1
2
x
1
+
6
x
2
+
5
x
3
+
2
x
4
=
0
→
swap
(
ρ
1
,
ρ
2
)
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
=
1
x
3
+
2
x
4
=
3
2
x
1
+
6
x
2
+
5
x
3
+
2
x
4
=
0
(
0
1
0
1
0
0
0
0
1
)
(
0
0
1
2
3
1
3
3
2
1
2
6
5
2
0
)
=
(
1
3
3
2
1
0
0
1
2
3
2
6
5
2
0
)
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
=
1
x
3
+
2
x
4
=
3
2
x
1
+
6
x
2
+
5
x
3
+
2
x
4
=
0
→
ρ
3
′
=
ρ
3
-2
ρ
1
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
=
1
x
3
+
2
x
4
=
3
-
x
3
-
2
x
4
=
-2
(
1
0
0
0
1
0
-2
0
1
)
(
1
3
3
2
1
0
0
1
2
3
2
6
5
2
0
)
=
(
1
3
3
2
1
0
0
1
2
3
0
0
-1
-2
-2
)
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
=
1
x
3
+
2
x
4
=
3
-
x
3
-
2
x
4
=
-2
→
ρ
3
′
=
ρ
3
+
ρ
2
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
=
1
x
3
+
2
x
4
=
3
0
=
1
(
1
0
0
0
1
0
0
1
1
)
(
1
3
3
2
1
0
0
1
2
3
0
0
-1
-2
-2
)
=
(
1
3
3
2
1
0
0
1
2
3
0
0
0
0
1
)
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
=
1
x
3
+
2
x
4
=
3
0
=
1
→
ρ
1
′
=
ρ
1
-3
ρ
2
{
x
1
+
3
x
2
-
4
x
4
=
-8
x
3
+
2
x
4
=
3
0
=
1
(
1
-3
0
0
1
0
0
0
1
)
(
1
3
3
2
1
0
0
1
2
3
0
0
0
0
1
)
=
(
1
3
0
-4
-8
0
0
1
2
3
0
0
0
0
1
)
{
x
3
+
2
x
4
=
5
x
1
+
3
x
2
+
3
x
3
+
2
x
4
=
3
2
x
1
+
6
x
2
+
5
x
3
+
2
x
4
=
1
→
swap
(
ρ
1
,
ρ
2
)
{
x
1
+
3
x
2
+
3
x
3
+
2
x
4
=
3
x
3
+
2
x
4
=
5
2
x
1
+
6
x
2
+
5
x
3
+
2
x
4
=
1
(
0
1
0
1
0
0
0
0
1
)
(
0
0
1
2
5
1
3
3
2
3
2
6
5
2
1
)
=
(
1
3
0
-4
a
0
0
1
2
b
0
0
0
0
c
)
{
x
3
+
2
x
4
=
5
x
1
+
3
x
2
+
3
x
3
+
2
x
4
=
3
2
x
1
+
6
x
2
+
5
x
3
+
2
x
4
=
1
(
1
-3
0
0
1
0
0
0
1
)
(
1
0
0
0
1
0
0
1
1
)
(
1
0
0
0
1
0
-2
0
1
)
(
0
1
0
1
0
0
0
0
1
)
(
5
A
3
1
)
=
(
1
3
0
-4
a
0
0
1
2
b
0
0
0
0
c
)
(
-3
1
0
1
0
0
1
-2
1
)
(
0
0
1
2
5
1
3
3
2
3
2
6
5
2
1
)
=
(
1
3
0
-4
a
0
0
1
2
b
0
0
0
0
c
)
Gauss Method
2×2 inverse
{
5
x
1
+
4
x
2
=
1
9
x
1
+
7
x
2
=
0
{
5
x
1
+
4
x
2
=
0
9
x
1
+
7
x
2
=
1
(
1
5
0
0
1
)
(
5
4
1
0
9
7
0
1
)
=
(
1
4
5
1
5
0
9
7
0
1
)
(
1
5
0
0
1
)
(
5
4
1
0
9
7
0
1
)
=
(
1
4
5
1
5
0
9
7
0
1
)
(
1
0
-9
1
)
(
1
4
5
1
5
0
9
7
0
1
)
=
(
1
4
5
1
5
0
0
-1
5
-9
5
1
)
(
1
0
-9
1
)
(
1
5
0
0
1
)
(
5
4
1
0
9
7
0
1
)
=
(
1
4
5
1
5
0
0
-1
5
-9
5
1
)
(
1
0
0
-5
)
(
1
4
5
1
5
0
0
-1
5
-9
5
1
)
=
(
1
4
5
1
5
0
0
1
9
-5
)
(
1
0
0
-5
)
(
1
0
-9
1
)
(
1
5
0
0
1
)
(
5
4
1
0
9
7
0
1
)
=
(
1
4
5
1
5
0
0
1
9
-5
)
(
1
-4
5
0
1
)
(
1
4
5
1
5
0
0
1
9
-5
)
=
(
1
0
-7
4
0
1
9
-5
)
(
1
-4
5
0
1
)
(
1
0
0
-5
)
(
1
0
-9
1
)
(
1
5
0
0
1
)
⏟
B
(
5
4
1
0
9
7
0
1
)
⏟
(
A
|
I
)
=
(
1
0
7
-4
0
1
-9
5
)
⏟
(
B
A
|
B
I
)
Gauss Method
3×3 inverse
{
-
x
2
+
x
3
=
1
2
x
1
+
3
x
2
-
2
x
3
=
0
x
1
+
2
x
2
-
x
3
=
0
{
-
x
2
+
x
3
=
0
2
x
1
+
3
x
2
-
2
x
3
=
1
x
1
+
2
x
2
-
x
3
=
0
{
-
x
2
+
x
3
=
0
2
x
1
+
3
x
2
-
2
x
3
=
0
x
1
+
2
x
2
-
x
3
=
1
(
0
-1
1
1
0
0
2
3
-2
0
1
0
1
2
-1
0
0
1
)
(
0
-1
1
1
0
0
2
3
-2
0
1
0
1
2
-1
0
0
1
)
→
swap
(
ρ
1
,
ρ
3
)
(
1
2
-1
0
0
1
2
3
-2
0
1
0
0
-1
1
1
0
0
)
(
0
0
1
0
1
0
1
0
0
)
⏟
B
(
0
-1
1
1
0
0
2
3
-2
0
1
0
1
2
-1
0
0
1
)
⏟
(
A
|
I
)
=
(
1
2
-1
0
0
1
2
3
-2
0
1
0
0
-1
1
1
0
0
)
⏟
(
B
A
|
B
I
)
(
1
2
-1
0
0
1
2
3
-2
0
1
0
0
-1
1
1
0
0
)
→
ρ
2
′
=
ρ
2
-2
ρ
1
(
1
2
-1
0
0
1
0
-1
0
0
1
-2
0
-1
1
1
0
0
)
(
1
0
0
-2
1
0
0
0
1
)
E
1
⏟
B
(
0
-1
1
1
0
0
2
3
-2
0
1
0
1
2
-1
0
0
1
)
⏟
(
A
|
I
)
=
(
1
2
-1
0
0
1
0
-1
0
0
1
-2
0
-1
1
1
0
0
)
⏟
(
B
A
|
B
I
)
(
1
2
-1
0
0
1
0
-1
0
0
1
-2
0
-1
1
1
0
0
)
→
ρ
2
′
=
-
ρ
2
(
1
2
-1
0
0
1
0
1
0
0
-1
2
0
-1
1
1
0
0
)
(
1
0
0
0
-1
0
0
0
1
)
E
2
E
1
⏟
B
(
0
-1
1
1
0
0
2
3
-2
0
1
0
1
2
-1
0
0
1
)
⏟
(
A
|
I
)
=
(
1
2
-1
0
0
1
0
1
0
0
-1
2
0
-1
1
1
0
0
)
⏟
(
B
A
|
B
I
)
(
1
2
-1
0
0
1
0
1
0
0
-1
2
0
-1
1
1
0
0
)
→
ρ
3
′
=
ρ
3
+
ρ
2
(
1
2
-1
0
0
1
0
1
0
0
-1
2
0
0
1
1
-1
2
)
(
1
0
0
0
1
0
0
1
1
)
E
3
E
2
E
1
⏟
B
(
0
-1
1
1
0
0
2
3
-2
0
1
0
1
2
-1
0
0
1
)
⏟
(
A
|
I
)
=
(
1
2
-1
0
0
1
0
1
0
0
-1
2
0
0
1
1
-1
2
)
⏟
(
B
A
|
B
I
)
(
1
2
-1
0
0
1
0
1
0
0
-1
2
0
0
1
1
-1
2
)
→
ρ
1
′
=
ρ
1
+
ρ
3
(
1
2
0
1
-1
3
0
1
0
0
-1
2
0
0
1
1
-1
2
)
(
1
0
1
0
1
0
0
0
1
)
E
4
E
3
E
2
E
1
⏟
B
(
0
-1
1
1
0
0
2
3
-2
0
1
0
1
2
-1
0
0
1
)
⏟
(
A
|
I
)
=
(
1
2
0
1
-1
3
0
1
0
0
-1
2
0
0
1
1
-1
2
)
⏟
(
B
A
|
B
I
)
(
1
2
0
1
-1
3
0
1
0
0
-1
2
0
0
1
1
-1
2
)
→
ρ
1
′
=
ρ
1
-2
ρ
2
(
1
0
0
1
1
-1
0
1
0
0
-1
2
0
0
1
1
-1
2
)
(
1
-2
0
0
1
0
0
0
1
)
E
5
E
4
E
3
E
2
E
1
⏟
B
(
0
-1
1
1
0
0
2
3
-2
0
1
0
1
2
-1
0
0
1
)
⏟
(
A
|
I
)
=
(
1
0
0
1
1
-1
0
1
0
0
-1
2
0
0
1
1
-1
2
)
⏟
(
B
A
|
B
I
)
(
0
-1
1
1
0
0
2
3
-2
0
1
0
1
2
-1
0
0
1
)
→
(
1
0
0
1
1
-1
0
1
0
0
-1
2
0
0
1
1
-1
2
)
E
6
E
5
E
4
E
3
E
2
E
1
⏟
B
(
0
-1
1
1
0
0
2
3
-2
0
1
0
1
2
-1
0
0
1
)
⏟
(
A
|
I
)
=
(
1
0
0
1
1
-1
0
1
0
0
-1
2
0
0
1
1
-1
2
)
⏟
(
B
A
⏟
I
|
B
I
⏟
B
)