Calculul ecuatiilor matriciale

Fie A, BIMm× m ( C), A = a₁₁ a₁₂ a ₁₃ ..a_1m , B= b₁₁ b b₁₃ ..b_1m

a₂₁ a₂₂ a ₂₃ ..a_2m b₂₁ b b₂₃ ..b_2m

a₃₁ a₃₂ a ₃₃ ..a_3m b₃₁ b b₃₃ ..b_3m

......... ........

a_m1 a_m2 a _m3 ....a_mm b_m1b _m2 b_m3 ...b_mm

astfel incat A sa fie ne singulara ( deci sa existeA^-1). Sa consideram ecua-tiile matriceale :

AX=B, YA=B

inmultind prima ecuatie la stanga cu A^-1 si pe a doua la dreapta cu A^-1, se obtine:

A^-1(AX)= A^-1B, (YA) A^-1= BA^-1

folosind asociativitatea matricilor se obtine:

(A^-1A)X= A^-1B, Y(A A^-1)= BA^-1

dar (A^-1A)=I_m si folosind proprietatea matricii identice,se va obtine:

X= A^-1B, Y= BA^-1

iar prin calculul A^-1B si B A^-1 se va obtine X, Y. De obicei X,Y sunt diferite deoarece inmultirea matricilor nu este comutativa:

Sa se rezolve urmatoarele ecuatii matriciale:

2 1 5 6 2 1 5 6

· X = ,notez A= ,B=

2 3 6 8 2 3 6 8

A^-1·/A·X=B A^-1·(A·X)= A^-1B (A^-1·A)·X= A^-1B I₂·X= A^-1B X= A^-1B

detA = 2·3 - 2 = 4

A^-1=(1/ detA)·A*

a*₁₁ a*₂₁

A*=

a*₁₂ a*₂₂

a*₁₁ =(-1)² ·3= 3

a*₁₂ =(-1)³ ·2= -2

a*₂₁ =(-1)³ ·1= -1          

a*₂₂ =(-1)⁴ ·2= 2

3 -1

A*=

-2 2

3 -1 3/4 -1/4

A^-1=(1/4) ·            =

-2 2 -1/2 1/2

3/4 -1/4 5 6 9/4 5/2

X= · =

-1/2 1/2 6 8 1/2 1

2. -1 -2 1 2 -1 -2 1 2

X· = ,notez A= ,B=

5 8 -3 6 5 8 -3 6

X·A=B/·A^-1 (X·A)·A^-1=B·A^-1 X·(A·A^-1)=B ·A^-1 X ·I₂=B· A^-1 X= B ·A^-1

detA = -8 +10 =2

A^-1=(1/ detA)·A*

a*₁₁ a*₂₁

A*=

a*₁₂ a*₂₂

a*₁₁ =(-1)² ·8= 8

a*₁₂ =(-1)³ ·(-2)=2

a*₂₁ =(-1)³ ·5= -5          

a*₂₂ =(-1)⁴ ·(-1)= -1

8 2

A*=

-5 -1

8 2 4 1

A^-1=(1/2) ·            =

-5 -1 -5/2 -1/2

7 2 4 1 23 6

X= · =

-3 5 -5/2 -1/2 -49/4 -11/2

3. -1 2 4 6 8 13 -1 2 4 6 8 13

·X· = ,notez A= ,B= ,C =

-3 8 5 8 -4 -7 -3 8 5 8 -4 -7

A·X·B=C/·B^-1 (A·X·B)·B^-1=C·B^-1 (A·X)·(B·B^-1)=C·B^-1 (A·X)·I₂=C·B^-1 (A^-1)·/A·X=C·B^-1 A^-1(A·X)=A^-1·C·B^-1 (A^-1·A)·X=A^-1·C·B^-1 I₂·X=A¹·C· B^-1 X=A^-1·C·B^-1

detA = -8 +6 =-2

A^-1=(1/ detA)·A*

a*₁₁ a*₂₁

A*=

a*₁₂ a*₂₂

a*₁₁ =(-1)² ·8= 8

a*₁₂ =(-1)³ ·(-3)=3

a*₂₁ =(-1)³ ·2= -2          

a*₂₂ =(-1)⁴ ·(-1)= -1

8 -2

A*=

3 -1

8 -2 -4 1

A^-1=[1/-(2)] ·            =

3 -1 -3/2 1/2

detB= 32-30 =2

B^-1=(1/ detB)·B*

b*₁₁ b*₂₁

B*=

b*₁₂ b*₂₂

b*₁₁ =(-1)² ·8= 8

b*₁₂ =(-1)³ ·5=-5

b*₂₁ =(-1)³ ·6= -6          

b*₂₂ =(-1)⁴ ·4= 4

8 -6

B*=

-5 4

8 -6 4 -3

B^-1=(1/2) ·            =

-5 4 -5/2 2

-4 1 8 13 4 -3 -36 -59 4 -3 7/2 -10

X= · · = · =

-3/2 1/2 -4 -7 -5/2 2 -14 -23 -5/2 2 3/2 -4

4. 1 2 3 -1 5 3 1 2 3 -1 5 3

X· 0 1 2 = 2 1 -1 ,notez A= 0 1 2 ,B= 2 1 -1

-1 2 1 -3 4 -5 -1 2 1 -3 4 -5

X·A=B/·A^-1 (X·A)·A^-1=B·A^-1 X·(A·A^-1)=B ·A^-1 X ·I₂=B· A^-1 X= B ·A^-1

detA = 1- 4 + 3 - 4 =

A^-1=(1/ detA)·A*

a*₁₁ a*₂₁ a*₃₁

A*= a*₁₂ a*₂₂ a*₃₂

a*₁₃ a*₂₃ a*₃₃

a*₁₁ =(-1)² ·(-3)= -3

a*₁₂ =(-1)³ ·2=-2

a*₁₃ =(-1)⁴ ·1= 1

a*₂₁ =(-1)³ ·(-4)=4

a*₂₂ =(-1)⁴ ·4= 4

a*₂₃ =(-1)⁵ ·4= 4          

a*₃₁ =(-1)⁴ ·1= 1

a*₃₂ =(-1)⁵ ·2=-2

a*₃₃ =(-1)⁶ ·1= 1

-3 4 1

A*= -2 4 -2

1 -4 1

-3 4 1

A^-1=[1/(-4)] · -2 4 -2

1 -4 1

-1 5 3 -3 4 1 -4 4 -8 1 -1 2

X = [1/(-4)] · 2 1 -1 · -2 4 -2 = [1/(-4)]· -9 16 -1 = 9/4 -4 1/4

-3 4 -5 1 -4 1 -4 24 -16 1 -6 4

5. 1 1 1 1 1 2 3 4 1 1 1 1 1 2 3 4

0 1 1 1 · X = 0 1 2 3 ,notez A= 0 1 1 1 ,B= 0 1 2 3

0 0 1 1 0 0 1 2 0 0 1 1 0 0 1 2

0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1

A^-1·/A·X=B A^-1·(A·X)= A^-1B (A^-1·A)·X= A^-1B I₂·X= A^-1B X= A^-1B

1 1 1 1

detA = 0 1 1 1 = 1

0 0 1 1

0 0 0 1

A^-1=(1/ detA)·A*

a*₁₁ a*₂₁ a*₃₁ a*₄₁

A*= a*₁₂ a*₂₂ a*₃₂ a*₄₂

a*₁₃ a*₂₃ a*₃₃ a*₄₃

a*₁₄ a*₂₄ a*₃₄ a*₄₄

a*₁₁ =(-1)² ·1= 1 a*₂₁ =(-1)³ ·1=-1 a*₃₁ =(-1)⁴ ·0= 0 a*₄₁ =(-1)⁵ ·0= 0

a*₁₂ =(-1)³ ·0= 0 a*₂₂ =(-1)⁴ ·1= 1 a*₃₂ =(-1)⁵ ·1=-1 a*₄₂ =(-1)⁶ ·0= 0

a*₁₃ =(-1)⁴ ·0= 0 a*₂₃ =(-1)⁵ ·0= 0 a*₃₃ =(-1)⁶ ·1= 1 a*₄₃ =(-1)⁷ ·1=-1

a*₁₄ =(-1)⁵ ·0= 0 a*₂₄ =(-1)⁶ ·0= 0 a*₃₄ =(-1)⁷ ·0= 0 a*₄₄ =(-1)⁸ ·1= 1

1 -1 0 0

A*= 0 1 -1 0 = A^-1

0 0 1 -1

0 0 0 1

1 -1 0 0 1 2 3 4 1 1 1 1

X = 0 1 -1 0 · 0 1 2 3 = 0 1 1 1

0 0 1 -1 0 0 1 2 0 0 1 1

0 0 0 1 0 0 0 1 0 0 0 1

6. 2 2 3 1 2 -3 0 -1 -1 2 2 3 1 2 -3 0 -1 -1

1 -1 0 ·X· 0 1 2 = 0 1 1 ,notez A= 1 -1 0 ,B= 0 1 2 ,C = 0 1 1

-1 2 1 0 0 1 0 0 1 -1 2 1 0 0 1 0 0 1

A·X·B=C/·B^-1 (A·X·B)·B^-1=C·B^-1 (A·X)·(B·B^-1)=C·B^-1 (A·X)·I₂=C·B^-1 (A^-1)·/A·X=C·B^-1 A^-1(A·X)=A^-1·C·B^-1 (A^-1·A)·X=A^-1·C·B^-1 I₂·X=A¹·C· B^-1 X=A^-1·C·B^-1

detA = -2+6-3-2=-1

A^-1=(1/ detA)·A*

a*₁₁ a*₂₁ a*₃₁

A*= a*₁₂ a*₂₂ a*₃₂

a*₁₃ a*₂₃ a*₃₃

a*₁₁ =(-1)² ·(-1)= -1 a*₂₁ =(-1)³ ·(-4)= 4 a*₃₁ =(-1)⁴ ·3= 3

a*₁₂ =(-1)³ ·1=-1 a*₂₂ =(-1)⁴ ·5= 5 a*₃₂ =(-1)⁵ ·(-3)= 3

a*₁₃ =(-1)⁴ ·1= 1 a*₂₃ =(-1)⁵ ·6= -6 a*₃₃ =(-1)⁶ ·(-4)= -4

-1 4 3

A*= -1 5 3

1 -6 -4

-1 4 3 1 -4 -3

A^-1=[1/(-1)] · -1 5 3 = 1 -5 -3

1 -6 -4 -1 6 4

detB= 1=1

B^-1=(1/ detB)·B*

b*₁₁ b*₂₁ b*₃₁

B*= b*₁₂ b*₂₂ b*₃₂

b*₁₃ b*₂₃ b*₃₃

b*₁₁ =(-1)² ·1= 1 b*₂₁ =(-1)³ ·2= -2 b*₃₁ =(-1)⁴ ·7= 7

b*₁₂ =(-1)³ ·0= 0 b*₂₂ =(-1)⁴ ·1= 1 b*₃₂ =(-1)⁵ ·2= -2

b*₁₃ =(-1)⁴ ·0= 0 b*₂₃ =(-1)⁵ ·0= 0 b*₃₃ =(-1)⁶ ·1= 1

1 -2 7

B*= 0 1 -2 = B^-1

0 0 1

1 -4 -3 0 -1 -1 1 -2 7 0 -5 -8 1 -2 7 0 -5 2

X= 1 -5 -3 · 0 1 1 · 0 1 -2 = 0 -6 -9 · 0 1 -2 = 0 -6 3

-1 6 4 0 0 1 0 0 1 0 7 11 0 0 1 0 7 -3

7. 5 3 4 5 3 4

X· -6 -3 -5 = (3 2 1) ,notez A= -6 -3 -5 ,B= ( 3 2 1)

4 2 2 4 2 2

X·A=B/·A^-1 (X·A)·A^-1=B·A^-1 X·(A·A^-1)=B ·A^-1 X ·I₂=B· A^-1 X= B ·A^-1

detA = -30-48-60+48+50+36 = -4

A^-1=(1/ detA)·A*

a*₁₁ a*₂₁ a*₃₁

A*= a*₁₂ a*₂₂ a*₃₂

a*₁₃ a*₂₃ a*₃₃

a*₁₁ =(-1)² ·4= 4 a*₂₁ =(-1)³ ·(-2)= 2 a*₃₁ =(-1)⁴ ·(-3)= -3

a*₁₂ =(-1)³ ·8=-8 a*₂₂ =(-1)⁴ ·(-6)= -6 a*₃₂ =(-1)⁵ ·(-1)= 1

a*₁₃ =(-1)⁴ ·0= 0 a*₂₃ =(-1)⁵ ·(-2)= 2 a*₃₃ =(-1)⁶ ·3= 3

4 2 -3

A*= -8 -6 1

0 2 3

4 2 -3

X = [1/(-4)] · (3 2 1) · -8 -6 1 = [1/(-4)]·(-4 -4 -4) = ( 1 1 1)

0 2 3

8. 2 0 1 1 2 3 2 0 1 1 2 3

X· 4 s 2 = 0 1 2 s R ,notez A= 4 s 2 ,B= 0 1 2

1 1 1 -1 1 2 1 1 1 -1 1 2

X·A=B/·A^-1 (X·A)·A^-1=B·A^-1 X·(A·A^-1)=B ·A^-1 X ·I₂=B· A^-1 X= B ·A^-1

A^-1=(1/ detA)·A*

detA = 2s+4-s-4=s

Cazul I s=0 A^-1

Cazul II s A^-1 detA=s

A^-1=(1/ detA)·A*

a*₁₁ a*₂₁ a*₃₁

A*= a*₁₂ a*₂₂ a*₃₂

a*₁₃ a*₂₃ a*₃₃

a*₁₁ =(-1)² ·(s-2)= s-2

a*₁₂ =(-1)³ ·2=-2

a*₁₃ =(-1)⁴ ·(4-s)=4-s

a*₂₁ =(-1)³ ·(-1)=1

a*₂₂ =(-1)⁴ ·1= 1

a*₂₃ =(-1)⁵ ·2=-2          

a*₃₁ =(-1)⁴ ·(-s)=-s

a*₃₂ =(-1)⁵ ·0=0

a*₃₃ =(-1)⁶ ·2s=2s

s-2 1 -s

A*= -2 1 0

4-s -2 2s

1 2 3 s-2 1 -s 6-2s -3 5s (6-2s)/s -3/s 5

X = 1 / s · 0 1 2 · -2 1 0 =1/s · 2-2s -3 4s = (2-2s)/s -3/s 4

-1 1 2 4-s -2 2s 8-3s -4 5s (8-3s)/s -4/s 5

9. 1 2 3 6 9 8 1 2 3 6 9 8

X· 2 3 4 = ,notez A= 2 3 4 ,B=

3 4 1 0 1 6 3 4 1 0 1 6

X·A=B/·A^-1 (X·A)·A^-1=B·A^-1 X·(A·A^-1)=B ·A^-1 X ·I₂=B· A^-1 X= B ·A^-1

detA = 3+24+24-27-16-4 = 4

A^-1=(1/ detA)·A*

a*₁₁ a*₂₁ a*₃₁

A*= a*₁₂ a*₂₂ a*₃₂

a*₁₃ a*₂₃ a*₃₃

a*₁₁ =(-1)² ·(-13)= -13

a*₁₂ =(-1)³ ·(-10)=10

a*₁₃ =(-1)⁴ ·(-1)= -1

a*₂₁ =(-1)³ ·(-10)=10

a*₂₂ =(-1)⁴ ·(-8)= -8

a*₂₃ =(-1)⁵ ·(-2)= 2          

a*₃₁ =(-1)⁴ ·(-1)= -1

a*₃₂ =(-1)⁵ ·(-2)=2

a*₃₃ =(-1)⁶ ·(-1)= -1

-13 10 -1

A*= 10 -8 2

-1 2 -1

6 9 8 -13 10 -1 4 4 4 1 1 1

X = (1/4) · · 10 -8 2 = (1/-4)· =

0 1 6 -1 2 -1 4 4 -4 1 1 -1