Ejemplos matriz inversa

1. MATRICES INVERSAS

PROBLEMA 1


A=





A= -20+72+24-90-12+32 = 6


A11= (5 · -2)- (1 · 6) = -10-6 = 16



A12= - - [(4 · -2)- (3 · 6)] = 8 + 18 = 26



A13= (4 · 1)- (3 · 5)= 4 - 15 = -11



A21= - - [(4 · -2)- (1 · 6)] = 8 + 6 = 14



A22= (2 · -2)- (3 · 6) = -4 - 18 = -22



A23 = - - [(2 · 1)- (3 · 4)] = -2 + 12 = 10



A31= (4 · 6)- (5 · 6) = 24 - 30 = -6



A32 = - - [(2 · 6)- (4 · 6)] = -12 + 24 = 12









2 4 6
4 5 6
3 1 -2
2 4 6 2 4
4 5 6 4 5
3 1 -2 3 1
5 6
1 -2
4 6
3 -2
4 5
3 1
4 6
1 -2
2 6
3 -2
2 4
3 1
4 6
5 6
2 6
4 6

2. A
-1
=






-16 (2) + 14 (4) -6 (3) = 6
-16 (4) + 14 (5) -6 (1) = 0
-16 (6) + 14 (6) -6 (-2)= 0



-11 (2) + 10 (4) -6 (3) = 0
-11 (4) + 10 (5) -6 (1) = 0
-11 (6) + 10 (6) -6 (-2)= 6



26 (2) -22 (4) +12 (3) = 0
26 (4) -22 (5) +12 (1) = 6
26 (6) -22 (6) +12 (-2)= 0



A
-1
=








A
-1
=







PROBLEMA 2


A =







A= -3 + 2 - 6 - 1 = -8



A11 = (-1 · 1)- (1 · 2) = -1-2 = -3



A12 = - - [(1 · 1)- (1 · 2)] = -1 + 2 = 1



A13 = (-1 · 1)- (1 · 2) = -1-2 = -3


2 4 6
4 5 6
3 1 -2
-16 14 -6
26 -22 12
-11 10 -6
6 0 0
0 6 0
0 0 6
1 0 0
0 1 0
0 0 1
3 1 0
1 -1 2
1 1 1
3 1 0 3 1
1 -1 2 1 -1
1 1 1 1 1
-1 2
1 1
1 2
1 1
1 2
1 1

3. A21= - - [(1 · 1)- (1 · 0)] = -1 + 0 = -1



A22= (3 · 1)- (1 · 0) = 3 - 0 = 3



A23= - - [(3 · 1)- (1 · 1)] = -3 + 1 = -2



A31= (1 · 2)- (-1 · 0) = 2 - 0 = 2




A32= - - [(3 · 2)- (1 · 0)] = -6 - 0 = -6



A33= (3 · -1)- (1 · 1) = -3 - 1 = -4




A
-1
=






-3(3) -1 (1) + 2(1) = -8
-3(1) - 1(-1) + 2(1) = 0
-3(0) - 1(2) + 2(1)= 0

1(3) + 3(1) - 6(1) = 0
1(1) + 3(-1) - 6(1) = -8
1(0) + 3(2) - 6 (1)= 0

2(3) - 2(1) - 4(1) = 0
2(1) - 2(-1) - 4(1) = 0
2(0) - 2(2) - 4(1)= -8





A
-1
=







A
-1
=








1 0
1 1
3 0
1 1
3 1
1 1
1 0
-1 2
3 0
1 2
3 1
1 -1
-3 -1 2
1 3 -6
2 -2 -4
3 1 0
1 -1 2
1 1 1
-8 0 0
0 -8 0
0 0 -8
-1 0 0
0 -1 0
0 0 -1

4. PROBLEMA 3


A=






A= 2 + 3 - 2 - 4 = -1



A11= (1 · 2)- (1 · 2) = 2-2 = 0



A12 = - - [(1 · 2)- (0 · 2)] = -2 - 0 = -2



A13= (1 · 1)- (0 · 1) = 1- 0 = 1



A21= - - [(2 · 2)- (1 · 3)] = -4 + 3 = -1



A22= (1 · 2)- (0 · 3) = 2 - 0 = 2



A23= - - [(1 · 1)- (0 · 1)] = -1 - 0 = -1



A31= (2 · 2)- (1 · 3) = 4 - 3 = 1



A32= - - [(1 · 2)- (1 · 3)] = -2 +3 = 1



A33= (1 · 1)- (1 · 2) = 1 - 2 = -1








1 2 3
1 1 2
0 1 2
1 2 3 1 2
1 1 2 1 1
0 1 2 0 1
1 2
1 2
1 2
0 2
1 1
0 1
2 3
1 2
1 3
0 2
1 2
0 1
2 3
1 2
1 3
1 2
1 2
1 1

5. A
-1
=






-1 (1) + 1(0) = -1
- 1(1) + 1(1) = 0
- 1(2) + 1(2)= 0

-2(1) + 2 (1) + 1(0) = 0
-2(2) + 2(1) + 1(1) = -1
-2(3) + 2(2) + 1(2)= 0

1(1) - 1 (1) - 1(0) = 0
1(2) - 1(1) - 1(1) = 0
1(3) - 1(2) - 1(2)= -1





A
-1
= 1






A
-1
=

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