Ecuatii Matriciale

Calculul ecuatiilor matriciale

Fie A, BIMm× m ( C), A = a11 a12 a 13 ...a1m , B= b11 b 12 b13 b1m

a21 a22 a 23 ...a2m b21 b 22 b23 b2m

a31 a32 a 33 ...a3m b31 b 32 b33 b3m

am1 am2 a m3 ..amm bm1b m2 bm3 ..bmm 16345kkc86cdz2c

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)=Im 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:

1. 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 = -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?0?? 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

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