p2k2kk
Fie A, B?Mm× 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

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=a1/-(2)i • =
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=a1/(-4)i • -2 4 -2
1 -4 1

-1 5 3 -3 4 1 -4 4 -8 1 -1 2
X = a1/(-4)i • 2 1 -1 • -2 4 -2 = a1/(-4)i• -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=a1/(-1)i • -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 = a1/(-4)i • (3 2 1) • -8 -6 1 = a1/(-4)i•(-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