4元1次方程式

4元1次方程式の計算

■4元1次方程式

a 11 x 1 + a 12 x 2 + a 13 x 3 + a 14 x 4 = b 1   ・・・・・・(1)

a 21 x 1 + a 22 x 2 + a 23 x 3 + a 24 x 4 = b 2   ・・・・・・(2)

a 31 x 1 + a 32 x 2 + a 33 x 3 + a 34 x 4 = b 3   ・・・・・・(3)

a 41 x 1 + a 42 x 2 + a 43 x 3 + a 44 x 4 = b 4   ・・・・・・(4)

x 4 を消去する.

(1)× a 24

a 11 a 24 x 1 + a 12 a 24 x 2 + a 13 a 24 x 3 + a 14 a 24 x 4 = a 24 b 1   ・・・・・・(5)

(2)× a 14

a 14 a 21 x 1 + a 14 a 22 x 2 + a 14 a 23 x 3 + a 14 a 24 x 4 = a 14 b 2   ・・・・・・(6)

(5)−(6)

( a 11 a 24 a 14 a 21 ) x 1 +( a 12 a 24 a 14 a 22 ) x 2 +( a 13 a 24 a 14 a 23 ) x 3 = a 24 b 1 a 14 b 2   ・・・・・・(7)

(1)× a 34

a 11 a 34 x 1 + a 12 a 34 x 2 + a 13 a 34 x 3 + a 14 a 34 x 4 = a 34 b 1   ・・・・・・(8)

(3)× a 14

a 14 a 31 x 1 + a 14 a 32 x 2 + a 14 a 33 x 3 + a 14 a 34 x 4 = a 14 b 3  ・・・・・・(8)

(8)−(9)

( a 11 a 34 a 14 a 31 ) x 1 +( a 12 a 34 a 14 a 32 ) x 2 +( a 13 a 34 a 14 a 33 ) x 3 = a 34 b 1 a 14 b 3  ・・・・・・(10)

(1)× a 44

a 11 a 44 x 1 + a 12 a 44 x 2 + a 13 a 44 x 3 + a 14 a 44 x 4 = a 44 b 1  ・・・・・・(11)

(4)× a 14

a 14 a 41 x 1 + a 14 a 42 x 2 + a 14 a 43 x 3 + a 14 a 44 x 4 = a 14 b 4  ・・・・・・(12)

(11)−(12)

( a 11 a 44 a 14 a 41 ) x 1 +( a 12 a 44 a 14 a 42 ) x 2 +( a 13 a 44 a 14 a 43 ) x 3 = a 44 b 1 a 14 b 4  ・・・・・・(13)

(7),(10),(13)より x 3 を消去する.

(7)× ( a 13 a 34 a 14 a 33 )

( a 13 a 34 a 14 a 33 ) ( a 11 a 24 a 14 a 21 ) x 1 + ( a 13 a 34 a 14 a 33 ) ( a 12 a 24 a 14 a 22 ) x 2 + ( a 13 a 34 a 14 a 33 ) ( a 13 a 24 a 14 a 23 ) x 3 = ( a 13 a 34 a 14 a 33 ) ( a 24 b 1 a 14 b 2 )  ・・・・・・(14)

(10)× ( a 13 a 24 a 14 a 23 )

( a 13 a 24 a 14 a 23 ) ( a 11 a 34 a 14 a 31 ) x 1 + ( a 13 a 24 a 14 a 23 ) ( a 12 a 34 a 14 a 32 ) x 2 + ( a 13 a 24 a 14 a 23 ) ( a 13 a 34 a 14 a 33 ) x 3 = ( a 13 a 24 a 14 a 23 ) ( a 34 b 1 a 14 b 3 )  ・・・・・・(15)

(14)−(15)

{ ( a 13 a 34 a 14 a 33 ) ( a 11 a 24 a 14 a 21 ) ( a 13 a 24 a 14 a 23 ) ( a 11 a 34 a 14 a 31 ) } x 1 + { ( a 13 a 34 a 14 a 33 ) ( a 12 a 24 a 14 a 22 ) ( a 13 a 24 a 14 a 23 ) ( a 12 a 34 a 14 a 32 ) } x 2 = ( a 13 a 34 a 14 a 33 ) ( a 24 b 1 a 14 b 2 ) ( a 13 a 24 a 14 a 23 ) ( a 34 b 1 a 14 b 3 )  ・・・・・・(16)

(16)の x 1 の係数を整理する.

( a 11 a 24 a 14 a 21 )( a 13 a 34 a 14 a 33 )( a 13 a 24 a 14 a 23 )( a 11 a 34 a 14 a 31 )

= a 11 a 24 a 13 a 34 a 11 a 24 a 14 a 33 a 14 a 21 a 13 a 34 + a 14 a 21 a 14 a 33 a 13 a 24 a 11 a 34 + a 13 a 24 a 14 a 31 + a 14 a 23 a 11 a 34 a 14 a 23 a 14 a 31

= a 11 a 13 a 24 a 34 a 11 a 14 a 24 a 33 a 13 a 14 a 21 a 34 + a 14 a 14 a 21 a 33 a 11 a 13 a 24 a 34 + a 13 a 14 a 24 a 31 + a 11 a 14 a 23 a 34 a 14 a 14 a 23 a 31

= a 11 a 14 a 24 a 33 a 13 a 14 a 21 a 34 + a 14 a 14 a 21 a 33 + a 13 a 14 a 24 a 31 + a 11 a 14 a 23 a 34 a 14 a 14 a 23 a 31

= a 14 ( a 11 a 24 a 33 a 13 a 21 a 34 + a 14 a 21 a 33 + a 13 a 24 a 31 + a 11 a 23 a 34 a 14 a 23 a 31 )

= A 11

とおく.

(16)の x 2 の係数を整理する.

( a 12 a 24 a 14 a 22 )( a 13 a 34 a 14 a 33 )( a 13 a 24 a 14 a 23 )( a 12 a 34 a 14 a 32 )

= a 12 a 24 a 13 a 34 a 12 a 24 a 14 a 33 a 14 a 22 a 13 a 34 + a 14 a 22 a 14 a 33 a 13 a 24 a 12 a 34 + a 13 a 24 a 14 a 32 + a 14 a 23 a 12 a 34 a 14 a 23 a 14 a 32

= a 12 a 13 a 24 a 34 a 12 a 14 a 24 a 33 a 13 a 14 a 22 a 34 + a 14 a 14 a 22 a 33 a 12 a 13 a 24 a 34 + a 13 a 14 a 24 a 32 + a 12 a 14 a 23 a 34 a 14 a 14 a 23 a 32

= a 12 a 14 a 24 a 33 a 13 a 14 a 22 a 34 + a 14 a 14 a 22 a 33 + a 13 a 14 a 24 a 32 + a 12 a 14 a 23 a 34 a 14 a 14 a 23 a 32

= a 14 ( a 12 a 24 a 33 a 13 a 22 a 34 + a 14 a 22 a 33 + a 13 a 24 a 32 + a 12 a 23 a 34 a 14 a 23 a 32 )

= A 12

とおく.

(16)の定数項を整理する.

( a 24 b 1 a 14 b 2 )( a 13 a 34 a 14 a 33 )( a 13 a 24 a 14 a 23 )( a 34 b 1 a 14 b 3 )

= a 24 b 1 a 13 a 34 a 24 b 1 a 14 a 33 a 14 b 2 a 13 a 34 + a 14 b 2 a 14 a 33 a 13 a 24 a 34 b 1 + a 13 a 24 a 14 b 3 + a 14 a 23 a 34 b 1 a 14 a 23 a 14 b 3

= a 13 a 24 a 34 b 1 a 14 a 24 a 33 b 1 a 13 a 14 a 34 b 2 + a 14 a 14 a 33 b 2 a 13 a 24 a 34 b 1 + a 13 a 14 a 24 b 3 + a 14 a 23 a 34 b 1 a 14 a 14 a 23 b 3

= a 14 a 24 a 33 b 1 a 13 a 14 a 34 b 2 + a 14 a 14 a 33 b 2 + a 13 a 14 a 24 b 3 + a 14 a 23 a 34 b 1 a 14 a 14 a 23 b 3

= a 14 ( a 24 a 33 b 1 a 13 a 34 b 2 + a 14 a 33 b 2 + a 13 a 24 b 3 + a 23 a 34 b 1 a 14 a 23 b 3 )

= B 1

とおく.

(14)−(15)を A 11 x 1 + A 12 x 2 = B 1 と表す.

(7)× ( a 13 a 44 a 14 a 43 )

( a 11 a 24 a 14 a 21 ) ( a 13 a 44 a 14 a 43 ) x 1 + ( a 12 a 24 a 14 a 22 ) ( a 13 a 44 a 14 a 43 ) x 2 + ( a 13 a 24 a 14 a 23 ) ( a 13 a 44 a 14 a 43 ) x 3 = ( a 24 b 1 a 14 b 2 ) ( a 13 a 44 a 14 a 43 )  ・・・・・・(17)

(13)× ( a 13 a 24 a 14 a 23 )

( a 11 a 44 a 14 a 41 ) ( a 13 a 24 a 14 a 23 ) x 1 + ( a 12 a 44 a 14 a 42 ) ( a 13 a 24 a 14 a 23 ) x 2 + ( a 13 a 44 a 14 a 43 ) ( a 13 a 24 a 14 a 23 ) x 3 = ( a 44 b 1 a 14 b 4 ) ( a 13 a 24 a 14 a 23 )  ・・・・・・(18)

(17)−(18)

{ ( a 11 a 24 a 14 a 21 ) ( a 13 a 44 a 14 a 43 ) ( a 11 a 44 a 14 a 41 ) ( a 13 a 24 a 14 a 23 ) } x 1 + { ( a 12 a 24 a 14 a 22 ) ( a 13 a 44 a 14 a 43 ) ( a 12 a 44 a 14 a 42 ) ( a 13 a 24 a 14 a 23 ) } x 2 = ( a 24 b 1 a 14 b 2 ) ( a 13 a 44 a 14 a 43 ) ( a 44 b 1 a 14 b 4 ) ( a 13 a 24 a 14 a 23 )  ・・・・・・(19)

(19)の x 1 の係数を整理する.

( a 11 a 24 a 14 a 21 ) ( a 13 a 44 a 14 a 43 ) ( a 13 a 24 a 14 a 23 ) ( a 11 a 44 a 14 a 41 ) = a 11 a 24 a 13 a 44 a 11 a 24 a 14 a 43 a 14 a 21 a 13 a 44 + a 14 a 21 a 14 a 43   a 13 a 24 a 11 a 44 + a 13 a 24 a 14 a 41 + a 14 a 23 a 11 a 44 a 14 a 23 a 14 a 41 = a 11 a 14 a 24 a 43 a 13 a 14 a 21 a 44 + a 14 a 14 a 21 a 43   + a 13 a 14 a 24 a 41 + a 11 a 14 a 23 a 44 a 14 a 14 a 23 a 41 = a 14 a 11 a 24 a 43 a 13 a 21 a 44 + a 14 a 21 a 43 + a 13 a 24 a 41 + a 11 a 23 a 44 a 14 a 23 a 41 = A 21

(19)の x 2 の係数を整理する.

( a 12 a 24 a 14 a 22 ) ( a 13 a 44 a 14 a 43 ) ( a 13 a 24 a 14 a 23 ) ( a 12 a 44 a 14 a 42 ) = a 12 a 24 a 13 a 44 a 12 a 24 a 14 a 43 a 14 a 22 a 13 a 44 + a 14 a 22 a 14 a 43   a 13 a 24 a 12 a 44 + a 13 a 24 a 14 a 42 + a 14 a 23 a 12 a 44 a 14 a 23 a 14 a 42 = a 12 a 14 a 24 a 43 a 13 a 14 a 22 a 44 + a 14 a 14 a 22 a 43   + a 13 a 14 a 24 a 42 + a 12 a 14 a 23 a 44 a 14 a 14 a 23 a 42 = a 14 a 12 a 24 a 43 a 13 a 22 a 44 + a 14 a 22 a 43 + a 13 a 24 a 42 + a 12 a 23 a 44 a 14 a 23 a 42 = A 22

(19)の定数項を整理する.

( a 24 b 1 a 14 b 2 ) ( a 13 a 44 a 14 a 43 ) ( a 44 b 1 a 14 b 4 ) ( a 13 a 24 a 14 a 23 ) = a 24 b 1 a 13 a 44 a 24 b 1 a 14 a 43 a 14 b 2 a 13 a 44 + a 14 b 2 a 14 a 43   a 44 b 1 a 13 a 24 + a 44 b 1 a 14 a 23 + a 14 b 4 a 13 a 24 a 14 b 4 a 14 a 23 = a 14 a 24 a 43 b 1 a 13 a 14 a 44 b 2 + a 14 a 14 a 43 b 2   + a 14 a 23 a 44 b 1 + a 13 a 14 a 24 b 4 a 14 a 14 a 23 b 4 = a 14 a 24 a 43 b 1 a 13 a 44 b 2 + a 14 a 43 b 2 + a 23 a 44 b 1 + a 13 a 24 b 4 a 14 a 23 b 4

= B 2

とおく.

(17)−(18)を

A 21 x 1 + A 22 x 2 = B 2

と表す.

A 11 x 1 + A 12 x 2 = B 1  ・・・・・・(20)

A 21 x 1 + A 22 x 2 = B 2  ・・・・・・(21)

X を消去する.

(20)× A 22

A 11 A 22 x 1 + A 12 A 22 x 2 = B 1 A 22  ・・・・・・(22)

(21)× A 12

A 12 A 21 x 1 + A 12 A 22 x 2 = A 12 B 2  ・・・・・・(23)

(22)−(23)

( A 11 A 22 A 12 A 21 ) x 1 = B 1 A 22 A 12 B 2

A 11 A 22 A 12 A 21

= { a 14 a 11 a 23 a 34 a 11 a 24 a 33 + a 13 a 24 a 31 a 13 a 21 a 34 + a 14 a 21 a 33 a 14 a 23 a 31   × a 14 a 12 a 23 a 44 a 12 a 24 a 43 + a 13 a 24 a 42 a 13 a 22 a 44 + a 14 a 22 a 43 a 14 a 23 a 42 }   { a 14 ( a 12 a 23 a 34 a 12 a 24 a 33 + a 13 a 24 a 32 a 13 a 22 a 34 + a 14 a 22 a 33 a 14 a 23 a 32 )   × a 14 a 11 a 23 a 44 a 11 a 24 a 43 + a 13 a 24 a 41 a 13 a 21 a 44 + a 14 a 21 a 43 a 14 a 23 a 41

= a 14 2 a 13 a 24 a 14 a 23 a 11 a 22 a 33 a 44 a 11 a 22 a 34 a 43 a 11 a 23 a 32 a 44 + a 11 a 23 a 34 a 42

+ a 11 a 24 a 32 a 43 a 11 a 24 a 33 a 42 a 12 a 21 a 33 a 44 + a 12 a 21 a 34 a 43 + a 12 a 23 a 31 a 44

a 12 a 23 a 34 a 41 a 12 a 24 a 31 a 43 + a 12 a 24 a 33 a 41 + a 13 a 21 a 32 a 44 a 13 a 21 a 34 a 42

a 13 a 22 a 31 a 44 + a 13 a 22 a 34 a 41 + a 13 a 24 a 31 a 42 a 13 a 24 a 32 a 41 a 14 a 21 a 32 a 43

+ a 14 a 21 a 33 a 42 + a 14 a 22 a 31 a 43 a 14 a 22 a 33 a 41 a 14 a 23 a 31 a 42 + a 14 a 23 a 32 a 41

= a 14 2 a 13 a 24 a 14 a 23 C 1

とおく.

B 1 A 22 A 12 B 2

= { a 14 b 1 a 23 a 34 b 1 a 24 a 33 + a 13 a 24 b 3 a 13 b 2 a 34 + a 14 b 2 a 33 a 14 a 23 b 3   × a 14 a 12 a 23 a 44 a 12 a 24 a 43 + a 13 a 24 a 42 a 13 a 22 a 44 + a 14 a 22 a 43 a 14 a 23 a 42 }   { a 14 ( a 12 a 23 a 34 a 12 a 24 a 33 + a 13 a 24 a 32 a 13 a 22 a 34 + a 14 a 22 a 33 a 14 a 23 a 32 )   × a 14 b 1 a 23 a 44 b 1 a 24 a 43 + a 13 a 24 b 4 a 13 b 2 a 44 + a 14 b 2 a 43 a 14 a 23 b 4

= a 14 2 a 13 a 24 a 14 a 23 a 22 a 33 a 44 b 1 a 22 a 34 a 43 b 1 a 23 a 32 a 44 b 1 + a 23 a 34 a 42 b 1

+ a 24 a 32 a 43 b 1 a 24 a 33 a 42 b 1 a 12 a 33 a 44 b 2 + a 12 a 34 a 43 b 2 + a 13 a 32 a 44 b 2

a 13 a 34 a 42 b 2 a 14 a 32 a 43 b 2 + a 14 a 33 a 42 b 2 + a 12 a 23 a 44 b 3 a 12 a 24 a 43 b 3

a 13 a 22 a 44 b 3 + a 13 a 24 a 42 b 3 + a 14 a 22 a 43 b 3 a 14 a 23 a 42 b 3 a 12 a 23 a 34 b 4

+ a 12 a 24 a 33 b 4 + a 13 a 22 a 34 b 4 a 13 a 24 a 32 b 4 a 14 a 22 a 33 b 4 + a 14 a 23 a 32 b 4

= a 14 2 a 13 a 24 a 14 a 23 C 2

とおく.

x 1 = B 1 A 22 A 12 B 2 A 11 A 22 A 12 A 21

= a 14 2   a 13 a 24 a 14 a 23 C 2 a 14 2   a 13 a 24 a 14 a 23 C 1

= C 2 C 1

 

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最終更新日: 2022年10月18日