Operaciones Con Matrices
Operaciones Con Matrices
Operaciones Con Matrices
a) A – B
(−11 32)−(−12 −53 )=(−1−1−2(−1 ) 3−3
= −1
)( 0 = −1 0
2−(−5 ) −1+1 2+5 0 7 )( )
b) 2A – 3B
2 (−11 32 )−3(−12 −53 )=(−11 xx22 32 xx 22)−(−12 xx33 3 x3
−5 x 3 )
9 = 2−6 6−9
¿ ( 2 6 )−( 6
−2 4 −3 −15 (−2−(−3 ) 4−(−15 ) )
)
−4 −3 −4 −3
¿(
−2+3 4+15 ) ( 1 19 )
=
2
c) (A. B)
2 2
1 3 x 2 3 = 1 x 2+ 3 x −1
( ) 1 x 3+3 x (−5 )
(( )(
−1 2 −1 −5 )) ( −1 x 2+ 2 x (−1 ) −1 x 3+2 x (−5 ) )
2
(−1 ) x (−1 )+ (−12 ) x (−4 ) (−1 ) x (−12 ) + (−12 ) x (−13 )
¿ −1 −12 = ) ( = 49 168
(
−4 −13 (−4 ) x (−1 ) + (−13 ) x (−4 ) (−4 ) x (−12 ) + (−13 ) x (−13 ) )(
56 217 )
d) A·B
(−11 32) x (−12 −53 )=(−11 xx2+3 x (−1 )
2+ 2 x (−1 )
1 x 3+3 x (−5 )
)(= −1 −12
−1 x 3+2 x (−5 ) −4 −13 )
e) B·A
3 x 1 3 = 2 x 1+3 x (−1 ) 2 x 3+3 x 2
(−12 )(
−5 −1 2 )(
−1 x 1+(−5) x (−1 ) −1 x 3+(−5)x 2
= −1 12
4 −13 )( )
a) A – B
2 0 1 1 0 1 2−1 0−0 1−0 1 0 1
( 5 1 1 )( )(
3 0 0 − 1 2 1 = 3−1 0−2 0−1 = 2 −2 −1
1 1 0 5−1 1−1 1−0 4 0 1 )( )
b) 2A – 3B
2 0 1 1 0 1 2 x2 0 x2 1x 2 1 x3 0 x3 1x 3
( ) ( )( )(
2 3 0 0 −3 1 2 1 = 3 x 2 0 x 2 0 x 2 − 1 x 3 2 x 3 1 x 3 =¿
5 1 1 1 1 0 5 x2 1x 2 1x 2 1 x3 1 x3 0 x3 )
4 0 2 3 0 3 4−3 0−0 2−3 1 0 −1
( 6 0 0
10 2 2
−
)(
3 6 3
3 3 0
=
)(
6−3 0−6 0−3
10−3 2−3 2−0
= 3 −6 −3
7 −1 2 )( )
c) A·B
2 0 1 1 0 1
( )( )
3 0 0 x 1 2 1 =¿
5 1 1 1 1 0
7 1 2
( 13 1 2
5 0 1 )
MATRICES INVERSAS
1 3
A
1. 4 2
Det. (A) :a 11 x A 11 −a12 x A 12 = 1x2–(3x4) = -10
[ 14 32]=[2 ]=2
. A11
A [ 1 3 ]=[ 4 ] =4
12
4 2
Matriz de Menores:
M 11 1 3 =[ 2 ] =2
4 [ 2]
1 3
M 12
4 [ 2] = [ 4 ]=4 M= [ 23 41]
1 3
M 21
4 [ 2] =[ 3 ] =3
M 22 1 3 =[ 1 ]=1
4 [ 2]
Matriz de Complementos Algebraicos:
A11 =M 11 x(−1)1+1 =2 x (−1)1+1=2
A12=M 12 x (−1)1+2=4 x (−1)1 +2=−4 A= [−32 −41 ]
A21=M 21 x (−1)2+1=3 x (−1)2+1=−3
A22=M 22 x (−1)2+2=1 x (−1)2+2=1
Matriz Traspuesta de Complementos Algebraicos:
T
A = [−42 −31 ]
1
Matriz Inversa: A =A T x ( )
−1
d
A−111 = AT 11 x ( d1 ) ( )
=2 x
1
−10
=−0.2
1 1
A−112= A T 12 x( )
=−3 x ( )=0.3
d −10
1 1
A−121= A T 21 x( )
=−4 x ( )=0.4
d −10
1 1
A−122= A T 22 x( )
=1 x ( ) =−0.1
d −10
RPTA:
0.3
−1
A = [−0.2
0.4 −0.1 ]
2 4
B
2. 3 1
Det. (B) :b 11 x B 11−b 12 x B12 = 2x1–(-4x-3) = -10
[−32 −41 ]= [1]=1
. B11
B [ 2 −4 ]=[ −3 ] =−3
12
−3 1
Matriz de Menores:
M 11 2 −4 = [ 1 ] =1
−3 [ 1 ]
M 12 2 M = 1 −3
−4 =[ −3 ] =−3
−3 [ 1 ] [
−4 2 ]
2 −4
M 21
−3 [ 1 ]
=[ −4 ] =−4
M 22 2 −4 =[ 2 ] =2
−3 [ 1 ]
Matriz de Complementos Algebraicos:
B11=M 11 x (−1)1+1=1 x (−1)1 +1=1
B12=M 12 x (−1)1 +2=−3 x (−1)1 +2=3 B= [ 14 32]
B21=M 21 x (−1)2 +1=−4 x (−1)2 +1=4
B22=M 22 x (−1)2 +2=2 x(−1)2+ 2=2
Matriz Traspuesta de Complementos Algebraicos:
T
B = [ 13 42 ]
1
Matriz Inversa: B =BT x ( )
−1
d
B−111 =BT 11 x ( 1d )=1 x (1
−10
=−0.1)
1 1
B−112=BT 12 x ( ) =4 x ( =−0.4)
d −10
1 1
B−121=BT 21 x ( ) =3 x ( =−0.3)
d −10
1 1
B−122=BT 22 x ( ) =2 x ( =−0.2)
d −10
RPTA:
B−1= −0.1 −0.4
[
−0.3 −0.2 ]
1 −2 3
3.
A= 5 −1 2
3 4 −3[ ]
Det. (A) :a 11 x A 11 −a12 x A 12+ a13 x A 13 = (1x-5)–(-2x-21)+(3x23) = 22
1 −2 3
[
. A11 5
3
1
−1 2 =
4 −3
−2 3
][
−1 2 =−5
4 −3 ]
[
A12 5
1
3
−1 2 =
4 −3
−2 3
][5 2 =−21
3 −3 ]
A13 5
[3
−1 2 =
4 −3 ][
5 −1 =23
3 4 ]
Matriz de Menores:
1 −2 3
M 11 5
3
1
[ −1
4
−2
][
2 =
−3
3
−1 2 =−5
4 −3 ]
M 12 5
3
1
[ −1
4
−2
][
−3
3
2 =
5 2 =−21
3 −3 ]
M 13 5
3
1
[ −1
4
−2
][
2 =
−3
3
5 −1 =23
3 4 ]
M 21 5
3
1
[ −1
4
−2
][
2 =
−3
3
−2 3 =−6
4 −3 ]
−5 −21 23
M 22 5
3
1
[ −1
4
−2
][
−3
3
2 =
1 3 =−12
3 −3 ] [
M = −6 −12 10
−1 −13 9 ]
M 23 5
3
1
[ −1
4
−2
][
2 =
−3
3
1 −2 =10
3 4 ]
M 31 5
3[ −1
4 ][
2 =
−3
−2 3 =−1
−1 2 ]
1 −2 3
M 32 5
3
1
[ −1 2 =
4 −3
−2 3
1
5 ] [ 32]=−13
[
M 33 5
3
−1 2 =
4 −3
1
5 ][ −2 =9
−1 ]
[
A= 6 −12 −10
−1 13 9 ]
A31=M 31 x (−1)3+1=−1 x (−1)3+1=−1
A32=M 32 x (−1)3+2=−13 x (−1)3+2=13
A33=M 33 x (−1)3+3 =9 x(−1)3 +3=9
[
A = 21 −12 13
23 −10 9 ] 1
Matriz Inversa: A =A T x ( )
−1
d
A−111 = AT 11 x ( d1 )
=−5 x
1
( )
22
=−0.227273
1 1
A−112= A T 12 x( ) =6 x ( )=0.272727
d 22
1 1
A−113= A T 13 x( ) =−1 x ( ) =−0.045455
d 22
1 1
A−121= A T 21 x( ) =21 x ( ) =0.954545
d 22
1 1
A−122= A T 22 x( ) =−12 x ( ) =0.545455
d 22
1 1
A−123= A T 23 x( ) =13 x ( )=0.590909
d 22
A−131= A T 31 x ( 1d )=23 x ( 221 )=1.045455
1 1
A−132= A T 32 x ( )=−10 x ( )=−0.454545
d 22
1 1
A−133= A T 33 x ( ) =9 x ( ) =−0.409091
d 22
RPTA:
−0.227273 0.272727 −0.045455
−1
[
A = 0.954545 −0.545455 0.590909
1.045455 −0.454545 0.409091 ]
−5 21 17
4.
[
B= −6 −12 2
1 13 9 ]
Det. (B) :b 11 x B 11−b 12 x B12+ b13 x B 13 = (-5x-134)–(21x-56)+(17x66) = 724
−5 21 17
[
. B11 −6
1
−5
−12
13
21
][
2 =
9
17
−12 2 =−134
13 9 ]
−5
[
B12 −6
1
−12
13
21
][
17
2 =
9
−6 2 =−56
1 9 ]
B13 −6
[
1
−12
13 ][
2 =
9
−6 −12 =−66
1 13 ]
Matriz de Menores:
−5 21 17
M 11 −6
1
−5
[ −12
13
21
][
2 =
9
17
−12 2 =−134
13 9 ]
M 12 −6
−5
1 [ −12
13
21 17
][
2 =
9
−6 2 =−56
1 9 ]
M 13 −6
1
−5
[ −12
13
21
][
2 =
9
17
−6 −12 =−66
1 13 ]
M 21 −6
1 [ −12
13 ][
2 =
9
21 17 =−32
13 9 ]
−5 21 17 −134 −56 −66
[
M 22 −6
1
−5
−12
13
21
9
17
][
2 =
−5
1
17 =−62
9] 246 [
M = −32 −62 −86
92 186 ]
[
M 23 −6
1
−5
−12
13
21
9
17
][
2 =
−5
1
21 =−86
13]
[
M 31 −6
1
−5
−12
13
21
9
17
][
2 =
21
−12
17 =246
2 ]
[
M 32 −6
1
−5
−12
13
21
9
17
][
2 =
−5
−6
17 =92
2]
[
M 33 −6
1
−12
13 ][
2 =
9
−5
−6
21 =186
−12 ]
[
BT = 56
−66
−62 −92
86 186 ] 1
Matriz Inversa: B =BT x ( )
−1
d
1 1
−1 T
B 11 =B 11 x()
d
=−134 x
724 ( )
=−0.185083
B−112=BT 12 x ( 1d )=32 x ( 7241 )=0.044199
1 1
x ( ) =246 x (
724 )
B−113=BT 13 =0.339779
d
1 1
x ( ) =56 x (
724 )
B−121=BT 21 =0.077348
d
1 1
x ( ) =−62 x (
724 )
B−122=BT 22 =−0.085635
d
1 1
x ( )=−92 x (
724 )
B−123=BT 23 =−0.127072
d
1 1
x ( ) =−66 x (
724 )
B−131=BT 31 =−0.09116
d
1 1
x ( ) =86 x (
724 )
B−132=BT 32 =0.118785
d
1 1
x ( ) =186 x (
724 )
B−133=BT 33 =0.256906
d
RPTA:
−0.185083 0.044199 0.339779
−1
[
B = 0.077348 −0.085635 −0.127072
−0.09116 0.118785 0.256906 ]
A= −2 3
5) 1 −5[ ]
Det. (A) :a 11 x A 11 −a12 x A 12 = -2x-5–(3x1) = 7
[−21 −53 ]=[−5]=−5
. A11
A [−2 3 ]=[ 1 ] =1
12
1 −5
Matriz de Menores:
3 = [−5 ] =−5
M 11 −2
1 [ −5 ]
−2 3 −5 1
M 12
1 [ −5 ]=[ 1 ] =1 M= [
3 −2 ]
−2 3
M 21
1 [ −5 ]=[ 3 ] =3
3 =[ −2 ] =−2
M 22 −2
1 [ −5 ]
Matriz de Complementos Algebraicos:
A11 =M 11 x(−1)1+1 =−5 x (−1 )1+1=−5
A= −5 −1
A12=M 12 x (−1)1+2=1 x (−1)1 +2=−1
2+1 2+1
[−3 −2 ]
A21=M 21 x (−1) =3 x (−1) =−3
A22=M 22 x (−1)2+2=−2 x (−1 )2+2=−2
Matriz Traspuesta de Complementos Algebraicos:
T
A = [−5 −3
−1 −2 ]
1
Matriz Inversa: A =A T x ( )
−1
d
A−111 = AT 11 x ( d1 )=−5 x( 1
7 )
=−0.714286
1 1
A−112= A T 12 x ( )=−3 x ( )
=−0.428571
d 7
1 1
A−121= A T 21 x ( )=−1 x ( )
=−0.142857
d 7
1 1
A−122= A T 22 x ( )=−2 x ( )
=−0.285714
d 7
RPTA:
−1
A = [−0.714286
−0.142857
−0.428571
−0.285714 ]
A= 1 −3
6)
[ ]
2 4
Matriz de Menores:
[ 12 −34 ]=[ 4]=4
M 11
M [1 −3 ]=[ 2 ] =2 M= 4 2
12
2 4 [
−3 1 ]
M [1 −3 ]=[−3 ] =−3
21
2 4
M 22 1 −3 =[ 1 ]=1
[
2 4 ]
Matriz de Complementos Algebraicos:
A11 =M 11 x(−1)1+1 =4 x (−1 )1+1=4
A= 4 −2
A12=M 12 x (−1)1+2=2 x (−1)1 +2=−2
2+1 2+1
[3 1 ]
A21=M 21 x (−1) =−3 x (−1) =3
A22=M 22 x (−1)2+2=1 x (−1 )2 +2=1
Matriz Traspuesta de Complementos Algebraicos:
T
A = [−24 31]
1
Matriz Inversa: A =A T x ( )
−1
d
A−111 = AT 11 x ( d1 )=4 x ( 101 )=0.4
1 1
A−112= A T 12 x ( )=3 x ( )=0.3
d 10
1 1
A−121= A T 21 x ( )=−2 x ( ) =−0.2
d 10
1 1
A−122= A T 22 x ( )=1 x ( )=0.1
d 10
RPTA:
0.4 0.3
−1
A = [−0.2 0.1 ]
A= 3 4
7. 2 5 [ ]
Det. (A) :a 11 x A 11 −a12 x A 12 = 3x5–(4x2) = 7
[32 45 ]=[5 ]=5
. A11
A [ 3 4 ]=[ 2 ] =2
12
2 5
Matriz de Menores:
M 11 3 4 =[ 5 ] =5
2 [ 5]
3 4
M 12
2 [ 5] = [ 2 ]=2 M= [ 54 23]
3 4
M 21
2 [ 5] =[ 4 ] =4
M 22 3 4 =[ 3 ] =3
2 [ 5]
Matriz de Complementos Algebraicos:
A11 =M 11 x(−1)1+1 =5 x (−1 )1+1=5
A= 5 −2
A12=M 12 x (−1)1+2=2 x (−1)1 +2=−2
2+1 2+1
[−4 3 ]
A21=M 21 x (−1) =4 x (−1) =−4
A22=M 22 x (−1)2+2=3 x (−1 )2+2=3
Matriz Traspuesta de Complementos Algebraicos:
A =
T
[−25 −43 ]
1
Matriz Inversa: A =A T x ( )
−1
d
A−111 = AT 11 x ( d1 )=5 x ( 17 )=0.714286
1 1
A−112= A T 12 x ( )=−4 x ( ) =−0.571429
d 7
1 1
A−121= A T 21 x ( )=−2 x ( ) =−0.285714
d 7
1 1
A−122= A T 22 x ( )=3 x ( ) =0.428571
d 7
RPTA:
0.714286 −0.571429
A =
−1
[−0.285714 0.428571 ]
−2 3 1
8.
[
A= 5 4 −3
1 3 6 ]
Det. (A) :a 11 x A 11 −a12 x A 12+ a13 x A 13 = (-2x33)–(3x33)+(1x11)= -154
−2 3 1
[
. A11 5
1
−2
][
4 −3 =
3 6
3 1
4 −3 =33
3 6 ]
[
A12 5
−2
1 ][
4 −3 =
3 6
3 1
5 −3 =33
1 6 ]
A13 5
[1 ][
4 −3 =
3 6
5 4 =11
1 3 ]
Matriz de Menores:
−2 3 1
M 11 5
1[
−2
4
3
3
][
−3 =
6
1
4 −3 =33
3 6 ]
M 12 5
−2
1[ 4
3
3
][
−3 =
6
1
5 −3 =33
1 6 ]
M 13 5
1
−2
[ 4
3
3
][
−3 =
6
1
5 4 =11
1 3 ]
M 21 5
1[
−2
4
3
3
][
−3 =
6
1
3 1 =15
3 6 ]
33 33 11
M 22 5
1
−2
[ 4
3
3
][
−3 =
6
1
−2 1 =−13
1 6 ] [
M = 15 −13 −9
−13 1 −23 ]
M 23 5
−2
[
1
4
3
3
][
−3 =
6
1
−2 3 =−9
1 3 ]
M 31 5
1[
−2
4
3
3
][
−3 =
6
1
3 1 =−13
4 −3 ]
M 32 5
1[ 4
3 ][
−3 =
6
−2 1 =1
5 −3 ]
−2 3 1
M 33
[
5 4 −3 =
1 3 6
−2 3 =−23
5 4 ][ ]
11 9[
AT = −33 −13 −1
−23 ] 1
Matriz Inversa: A =A T x ( )
−1
d
A−111 = AT 11 x ( d1 )
=33 x
1
(
−154 )
=−0.214286
1 1
A−112= A T 12 x( ) =−15 x ( )=0.097403
d −154
1 1
A−113= A T 13 x( ) =−13 x ( )
=0.084416
d −154
1 1
A−121= A T 21 x( ) =−33 x ( )
=0.214286
d −154
1 1
A−122= A T 22 x( ) =−13 x ( )
=0.084416
d −154
1 1
A−123= A T 23 x( ) =−1 x ( )=0.006494
d −154
1 1
A−131= A T 31 x( ) =11 x ( )
=−0.071429
d −154
A−132= A T 32 x ( 1d )=9 x ( −154
1
)=−0.058442
1 1
x ( ) =−23 x (
−154 )
A−133= A T 33 =0.149351
d
RPTA:
−0.214286 0.097403 0.084416
[
A−1= 0.214286 0.084416 0.006494
−0.071429 −0.058442 0.149351 ]
1 3 −4
9.
A= 5 2 −1
9 −6 8[ ]
Det. (A) :a 11 x A 11 −a12 x A 12+ a13 x A 13 = (1x10)–(3x49)+(-4x-48)= 55
1 3 −4
A
[
. 11 5 2 −1 =
9 −6 8
1 3 −4
−6 8 ][
2 −1 =10
]
[
A12 5 2 −1 = 5 −1 =49
9 −6 8
1 3 −4
9 8 ][ ]
[
A13 5 2 −1 = 5 2 =−48
9 −6 8
9 −6 ][ ]
Matriz de Menores:
1 3 −4
M 11 5
9
1
[ 2
−6
3
8
−4
][
−1 =
2 −1 =10
−6 8 ]
M 12 5
1
9[ 2
−6
3
8
−4
][
−1 =
5 −1 =49
9 8 ]
M 13 5
9
1
[ 2
−6
3
8
−4
][
−1 =
5 2 =−48
9 −6 ]
M 21 5
9[
1
2
−6
3
8
−4
][
−1 =
3 −4 =0
−6 8 ]
10 49 −48
M 22 5
9[ 2
−6 ][
−1 =
8
1 −4 =44
9 8 ] [
M = 0 44 −33
5 19 −13 ]
1 3 −4
[
M 23 5
9
1
2
−6
3
]
−1 =
8
−4
1
9 [ 3 =−33
−6 ]
[
M 31 5
9
1
2
−6
3
8
−4
][
−1 =
3
2
−4 =5
−1 ]
[
M 32 5
1
9
2
−6
3
8
−4
][
−1 =
1
5
−4 =19
−1 ]
[
M 33 5
9
2
−6 ][
−1 =
8
1
5
3 =−13
2 ]
[
A= 0 44 33
5 −19 −13 ]
A31=M 31 x (−1)3+1=5 x (−1)3+1=5
A32=M 32 x (−1)3+2=19 x (−1)3+2=−19
A33=M 33 x (−1)3+3 =−13 x (−1 )3+3 =−13
[
AT = −49 44 −19
−48 33 −13 ] 1
Matriz Inversa: A =A T x ( )
−1
d
A−111 = AT 11 x ( d1 ) ( )
=10 x
1
55
=0.181818
1 1
A−112= A T 12 x( )=0 x ( )=0
d 55
1 1
A−113= A T 13 x( )=5 x( ) =0.090909
d 55
A−121= A T 21 x ( 1d )=−49 x ( 551 )=−0.890909
1 1
A−122= A T 22 x ( )=44 x ( )=0.8
d 55
1 1
A−123= A T 23 x ( ) =−19 x ( ) =−0.345455
d 55
1 1
A−131= A T 31 x ( )=−48 x ( )=−0.872727
d 55
1 1
A−132= A T 32 x ( )=33 x ( )=0.6
d 55
1 1
A−133= A T 33 x ( )=−13 x ( ) =−0.236364
d 55
RPTA:
0.181818 0 0.090909
[
A−1= −0.890909 0.8 −0.345455
−0.872727 0.6 −0.236364 ]
6 5 3
10.
A= 2 4 5
[ ]
1 2 −3
6
1 [ ][
4 5 =
2 −3
5 3
2 5 =−11
1 −3 ]
A13 2
1 [ ][
4 5 =
2 −3
2 4 =0
1 2 ]
Matriz de Menores:
6 5 3
M 11 2
1
6
[ ][
4 5 =
2 −3
5 3
4 5 =−22
2 −3 ]
M 12 2
1 [ ][
4 5 =
2 −3
2 5 =−11
1 −3 ]
6 5 3
[
M 13 2
1
6
4
2
5
−3
3
]
5 =[ 42]=0
2
1
[
M 21 2
1
6
4
2
5
−3][
5 =
3
5
2
3 =−21
−3 ]
−22 −11 0
[
M 22 2
6
1
5
4
2 −3
3
][
5 =
6
1
3 =−21
−3 ] [
M = −21 −21 7
13 24 14 ]
[
M 23 2
1
6
4
2
5
−3
3
][
5 =
6
1
5 =7
2]
[
M 31 2
1
6
4
2
5
−3][
5 =
3
5
4
3 =13
5]
[
M 32 2
1
6 5
4
2 −3
3
][
5 =
6
2
3 =24
5]
[
M 33 2
1
4
2 ][
5 =
−3
6
2
5 =14
4]
0 −7[
A = 11 −21 −24
14 ] 1
Matriz Inversa: A =A T x ( )
−1
d
A−111 = AT 11 x ( d1 ) ( )
=−22 x
1
−77
=0.285714
1 1
A−112= A T 12 x( ) ( )
=21 x =−0.272727
d −77
1 1
A−113= A T 13 x( ) ( )
=13 x =−0.168831
d −77
1 1
A−121= A T 21 x( ) ( )
=11 x =−0.142857
d −77
1 1
A−122= A T 22 x( ) =−21 x( ) =0.272727
d −77
1 1
A−123= A T 23 x( ) =−24 x( ) =0.311688
d −77
1 1
A−131= A T 31 x( ) ( )
=0 x =0
d −77
1 1
A−132= A T 32 x( ) =−7 x( ) =0.090909
d −77
1 1
A−133= A T 33 x( ) ( )
=14 x =−0.181818
d −77
RPTA:
0.285714 −0.272727 −0.168831
[
A−1= −0.142857 0.272727 −0.311688
−0 0.090909 −0.181818 ]
A= 3 5
10. 1 4 [ ]
Det. (A) :a 11 x A 11 −a12 x A 12 = (3x4)–(5x1) = 7
[31 54 ]=[ 4 ]=4
. A11
3 5
A [
1 4]
12 =[ 1 ] =1
Matriz de Menores:
M 11 [ 31 54 ]=[ 4 ]=4
M 12 3
1 [ 54 ]= [1 ]=1 M= 4 1
5 3[ ]
M 21 3
1 [ 54 ]=[5 ]=5
M 22
3
1 [ 54 ]=[3 ]=3
Matriz de Complementos Algebraicos:
A11 =M 11 x(−1)1+1 =4 x (−1 )1+1=4
B= 4 −1
11. −5 3 [ ]
Det. (B) :b 11 x B 11−b 12 x B12 = (4x3)–(-1x-5) = 7
. B11 [−54 −13 ]= [3 ]=3
B12 4 −1 =[ −5 ] =−5
[
−5 3 ]
Matriz de Menores:
M 11 4 −1 = [ 3 ]=3
[
−5 3 ]
4
M 12 [
−5
−1
3 ]
=[ −5 ] =−5 M= [−13 −54 ]
4 −1
M 21 [
−5 3 ]
=[ −1 ] =−1
M 22 4 −1 =[ 4 ] =4
−5 [ 3 ]
Matriz de Complementos Algebraicos:
B11=M 11 x (−1)1+1=3 x (−1 )1 +1=3
B= 3 5
B12=M 12 x (−1)1 +2=−5 x (−1)1 +2=5
2 +1 2+ 1
[ ]
1 4
B21=M 21 x (−1) =−1 x(−1) =1
B22=M 22 x (−1)2 +2=4 x (−1 )2+2 =4
Matriz Traspuesta de Complementos Algebraicos:
T
B = [ 35 14 ]
1
Matriz Inversa: B =BT x ( )
−1
d
B−111 =BT 11 x ( 1d )=3 x ( 17 )=0.428571
1 1
B−112=BT 12 x ( ) =1 x ( ) =0.142857
d 7
1 1
B−121=BT 21 x ( ) =5 x ( )=0.714286
d 7
1 1
B−122=BT 22 x ( ) =4 x ( ) =0.571429
d 7
RPTA:
−1
B = [ 0.428571
0.714286
0.142857
0.571429 ]
1 5
12. D=
[ ]
−2 3
Det. (D) :d 11 x D11 −d 12 x D12 = (1x3)–(5x-2) = 13
[−21 53]= [3 ]=3
. D 11
D [ 1 5 ]=[ −2 ] =−2
12
−2 3
Matriz de Menores:
M 11 1 5 = [ 3 ]=3
−2[ 3]
1 5
M 12
−2[ 3] =[−2 ] =−2 M= [ 35 −21 ]
1 5
M 21
−2[ 3] =[ 5 ] =5
M 22 1 5 =[ 1 ] =1
−2 [ 3]
Matriz de Complementos Algebraicos:
D 11 =M 11 x (−1)1+1=3 x (−1 )1+1=3
D= 3 2
D 12=M 12 x (−1)1+2=−2 x (−1)1 +2=2
2+1 2 +1
[
−5 1 ]
D 21=M 21 x (−1) =5 x (−1) =−5
D 22=M 22 x (−1)2+2=1 x (−1 )2+ 2=1
Matriz Traspuesta de Complementos Algebraicos:
3 −5
T
D = [ 2 1 ]
1
Matriz Inversa: D =D T x( )
−1
d
D −111 =D T 11 x ( 1d ) ( )
=3 x
1
13
=0.230769
1 1
D −112=D T 12 x( ) ( )
=−5 x =−0.384615
d 13
1 1
D −121=D T 21 x( ) ( )
=2 x =0.153846
d 13
1 1
D −122=D T 22 x( ) ( )
=1 x =0.076923
d 13
RPTA:
−1
D = [ 0.230769
0.153846
−0.384615
0.076923 ]
13. E=
[−2−3 −14 ]
Det. (E) :e 11 x E11 −e 12 x E 12 = (-2x4)–(-1x-3) = -11
−2 −1 =[ 4 ] =4
[−3
. E11
4 ]
−2 −1
E [
−3 4 ]
12 =[ −3 ]=−3
Matriz de Menores:
[−2 −1
−3 4 ]
M 11 = [ 4 ] =4
M [−2 −1 ] =[ −3 ] =−3 M = 4 −3
12
−3 4 [
−1 −2 ]
M [−2 −1 ] =[ −1 ] =−1
21
−3 4
−2 −1
M [
−3 4 ]
22 =[ −2 ] =−2
1 1
E−112=E T 12 x( ) ( )
=1 x =−0.090909
d −11
1 1
E−121=E T 21 x( ) ( )
=3 x =−0.272727
d −11
1 1
E−122=E T 22 x( ) ( )
=−2 x =0.181818
d −11
RPTA:
E−1= −0.363636 −0.090909
[
−0.272727 0.181818 ]
14. C=
[72 39 ]
Det. (C) :c 11 x C 11−c12 x C 12 = (2x9)–(3x7) = -3
.C 11
2
7[ 39]= [9 ]=9
C 12
2
7[ 39]=[7 ]=7
Matriz de Menores:
[ 27 39]=[ 9]=9
M 11
M [2 3 ] =[ 7 ] =7 M= 9 7
12
7 9 [ ]
3 2
M [ 2 3 ] =[ 3 ] =3
21
7 9
2 3
M [
7 9]
22 =[ 2 ] =2
1 1
C−112=CT 12 x( ) ( )
=−3 x =1
d −3
1 1
C−121=CT 21 x( ) ( )
=−7 x =2.333333
d −3
1 1
C−122=CT 22 x( ) ( )
=2 x =−0.666667
d −3
RPTA:
−3 1
−1
C = [2.333333 −0.666667 ]
2 −3
15. F=
[ −4 −6 ]
Det. (F) : f 11 x F11 −f 12 x F 12 = (2x-6)–(-3x-4) = -24
[−42 −3
. F 11
−6 ]
= [−6 ] =−6
2 −3
F [
−4 −6 ]
12 =[ −4 ] =−4
Matriz de Menores:
[−42 −3
M 11
−6 ]
=[−6 ] =−6
M [ 2 −3 ]=[ −4 ] =−4 M = −6 −4
12
−4 −6 [ −3 2 ]
M [ 2 −3 ]=[ −3 ] =−3
21
−4 −6
2 −3
M [
−4 −6 ]
22 =[ 2 ] =2
1 1
F−112=F T 12 x( ) ( )
=3 x =−0.125
d −24
F−121=F T 21 x ( 1d )=4 x( −241 )=−0.166667
1 1
x ( )=2 x (
−24 )
F−122=F T 22 =−0.083333
d
RPTA:
0.25 −0.125
−1
F = [−0.166667 −0.083333 ]
16. G=
[ −6 −12
3 6 ]
Det. (G) : g11 x G 11 −g12 x G 12 = (-6x6)–(-12x3) = 0
[−63
. F 11
−12 =[ 6 ] =6
6 ]
−6 −12
F [
12
3 6 ]
= [ 3 ] =3
Es imposible calcular la matriz inversa, la matriz es degenerada (el determinante de la
matriz es igual a cero)
17. H =
[83 14 ]
Det. (H) :h11 x H 11−h 12 x H 12 = (8x4)–(1x3) = 29
8 1
[3 4 ]
. H 11 = [ 4 ]=4
H [ 8 1 ]=[ 3 ] =3
12
3 4
Matriz de Menores:
M [ 8 1 ]= [ 4 ] =4
11
3 4
8 1
M [
3 4]
12 =[ 3 ] =3 M= [ 41 38]
M [ 8 1 ]=[ 1 ] =1
21
3 4
8 1
M [
3 4]
22 =[ 8 ] =8
Matriz de Complementos Algebraicos:
H 11 =M 11 x (−1)1+1=4 x (−1 )1 +1=4
H= 4 −3
H 12=M 12 x (−1)1 +2=3 x (−1)1 +2=−3
2+1 2+ 1
[
−1 8 ]
H 21=M 21 x (−1) =1 x(−1) =−1
H 22=M 22 x (−1)2+2=8 x (−1 )2+2=8
Matriz Traspuesta de Complementos Algebraicos:
T
H = [−34 −18 ]
−1 T 1
Matriz Inversa: H =H x ( d )
H −111 =H T 11 x ( 1d )=4 x ( 291 )=0.137931
1 1
H −112=H T 12 x ( ) =−1 x ( )=−0.034483
d 29
1 1
H −121=H T 21 x ( )=−3 x ( )=−0.103448
d 29
1 1
H −122=H T 22 x ( )=8 x ( )=0.275862
d 29
RPTA:
0.137931 −0.034483
−1
H = [−0.103448 0.275862 ]
1 −2 3
18.
[
A= 5 −1 2
3 4 −3 ]
Det. (A) :a 11 x A 11 −a12 x A 12+ a13 x A 13 = (1x-5)–(-2x-21)+(3x23) = 22
1 −2 3
A
. 11 5
3
1
[ −1 2 =
4 −3
−2 3
][
−1 2 =−5
4 −3 ]
A12 5
1
3[ −1 2 =
4 −3
−2 3
][
5 2 =−21
3 −3 ]
A13 5
3[ −1 2 =
4 −3
3 4 ][
5 −1 =23
]
Matriz de Menores:
1 −2 3
[
M 11 5 −1 2 = −1 2 =−5
3 4 −3
4 −3 ][ ]
1 −2 3
[
M 12 5
3
1
−1
4
−2
−3
3
][
2 =
5 2 =−21
3 −3 ]
[
M 13 5
3
1
−1
4
−2
−3
3
][
2 =
5 −1 =23
3 4 ]
[
M 21 5
3
1
−1
4
−2
−3
3
][
2 =
−2 3 =−6
4 −3 ]
−5 −21 23
[
M 22 5
3
1
−1
4
−2
−3
3
][
2 =
1 3 =−12
3 −3 ] [
M = −6 −12 10
−1 −13 9 ]
[
M 23 5
3
1
−1
4
−2
−3
3
][
2 =
1 −2 =10
3 4 ]
[
M 31 5
3
1
−1
4
−2
−3
3
][
2 =
−2 3 =−1
−1 2 ]
[
M 32 5
3
1
−1
4
−2
−3
3
][
2 =
1 3 =−13
5 2 ]
[
M 33 5
3
−1
4 ][
2 =
−3
1 −2 =9
5 −1 ]
[
AT = 21 −12 13
23 −10 9 ] 1
Matriz Inversa: A =A T x ( )
−1
d
A−111 = AT 11 x ( d1 ) ( )
=−5 x
1
22
=−0.227273
1 1
A−112= A T 12 x( )
=6 x ( )=0.272727
d 22
1 1
A−113= A T 13 x( )
=−1 x ( ) =−0.045455
d 22
1 1
A−121= A T 21 x( )
=21 x ( ) =0.954545
d 22
1 1
A−122= A T 22 x( )
=−12 x ( ) =0.545455
d 22
1 1
A−123= A T 23 x( )
=13 x ( )=0.590909
d 22
1 1
A−131= A T 31 x( )
=23 x ( )=1.045455
d 22
1 1
A−132= A T 32 x( )
=−10 x ( ) =−0.454545
d 22
1 1
A−133= A T 33 x( )
=9 x ( ) =−0.409091
d 22
RPTA:
−0.227273 0.272727 −0.045455
−1
[
A = 0.954545 −0.545455 0.590909
1.045455 −0.454545 0.409091 ]
−5 21 17
19.
1 13[
B= −6 −12 2
9 ]
Det. (B) :b 11 x B 11−b 12 x B12+ b13 x B 13 = (-5x-134)–(21x-56)+(17x66) = 724
−5 21 17
[
. B11 −6 −12 2 =
1 13 9
13 9 ][
−12 2 =−134
]
−5 21 17
[
B12 −6
−5
1
−12
13
21
][
17
2 =
9
−6 2 =−56
]
1 9
[
B13 −6
1
−12
13 ][
2 =
9
−6 −12 =−66
1 13 ]
Matriz de Menores:
−5 21 17
[
M 11 −6
1
−5
−12
13
21
][
2 =
9
17
−12 2 =−134
13 9 ]
[
M 12 −6
−5
1
−12
13
21 17
][
2 =
9
−6 2 =−56
]
1 9
[
M 13 −6
1
−5
−12
13
21
][
2 =
9
17
−6 −12 =−66
1 13 ]
[
M 21 −6
1
−5
−12
13
21
][
2 =
9
17
21 17 =−32
13 9]
−134 −56 −66
[
M 22 −6
−5
1
−12
13
21 17
][
2 =
9
−5 17 =−62
1 9 ] [
M = −32 −62 −86
246 92 186 ]
[
M 23 −6
1
−5
−12
13
21
][
2 =
9
17
−5 21 =−86
1 13 ]
[
M 31 −6
1
−5
−12
13
21
][
2 =
9
17
21 17 =246
−12 2 ]
[
M 32 −6
−5
1
−12
13
21 17
][
2 =
9
−5 17 =92
−6 2 ]
[
M 33 −6
1
−12
13 ][
2 =
9
−5 21 =186
−6 −12 ]
1 1
B−112=BT 12 x( ) =32 x ( )=0.044199
d 724
1 1
B−113=BT 13 x( ) =246 x( ) =0.339779
d 724
1 1
B−121=BT 21 x( ) =56 x( ) =0.077348
d 724
1 1
B−122=BT 22 x( ) =−62 x ( ) =−0.085635
d 724
1 1
B−123=BT 23 x( ) =−92 x ( ) =−0.127072
d 724
1 1
x( ) ( )
−1 T
B 31 =B 31 =−66 x =−0.09116
d 724
1 1
B−132=BT 32 x( ) =86 x( ) =0.118785
d 724
1 1
B−133=BT 33 x( ) =186 x( ) =0.256906
d 724
RPTA:
−0.185083 0.044199 0.339779
−1
[
B = 0.077348 −0.085635 −0.127072
−0.09116 0.118785 0.256906 ]
1 1 0
20. A=
[ ]
1 0 1
0 1 0
[
A12 1
1
0
1
0
1
0
][
1=
0
1
0
1 =0
0]
[
A13 1
0
0
1 ][
1=
0
1
0
0 =1
1]
Matriz de Menores:
1 1 0
[
M 11 1
0
1
0
1
1
0
0
][
1=
0
1
1 =−1
0]
[
M 12 1
0
1
0
1
1
1=
0
0
][
1
0
1 =0
0 ]
[
M 13 1
0
1
0
1
1
0
0
][
1=
1
0
0 =1
1]
[
M 21 1
0
1
0
1
1
0
0
][
1=
1
1
0 =0
0]
−1 0 1
[
M 22 1
1
0
0
1
1
1=
0
0
][
1
0
0 =0
0 ] [
M= 0 0 1
1 1 −1 ]
[
M 23 1
0
1
0
1
1
0
0
][
1=
1
0
1 =1
1]
[
M 31 1
0
1
0
1
1
0
0
][
1=
1
0
0 =1
1]
[
M 32 1
0
0
1
1=
0 ][
1
1
0 =1
1 ]
1 1 0
M 33
[ ][
1 0 1=
0 1 0
1 1 =−1
1 0 ]
[
A= 0 0 −1
1 −1 −1 ]
A31=M 31 x (−1)3+1=1 x (−1)3 +1=1
A32=M 32 x (−1)3+2=1 x (−1)3 +2=−1
A33=M 33 x (−1)3+3 =−1 x (−1)3+3 =−1
[
A = 0 0 1
−1 1 1 ] 1
Matriz Inversa: A =A T x ( )
−1
d
A−111 = AT 11 x ( d1 ) =−1 x
1
( )
−1
=1
1 1
A−112= A T 12 x( ) =0 x ( )=0
d −1
1 1
A−113= A T 13 x( ) =1 x ( ) =−1
d −1
1 1
A−121= A T 21 x( ) =0 x ( ) =0
d −1
1 1
A−122= A T 22 x( ) =0 x ( ) =0
d −1
1 1
A−123= A T 23 x( ) =−1 x ( )=1
d −1
1 1
A−131= A T 31 x( ) =1 x ( ) =−1
d −1
A−132= A T 32 x ( 1d )=−1 x ( −11 )=1
1 1
x ( )=−1 x (
−1 )
A−133= A T 33 =1
d
RPTA:
1 0 −1
A−1= 0 0 1
−1 1 1[ ]
−1 2 1
21. L=
[ 1 0 2
2 1 −1 ]
Det. (L) :l11 x L11 −l12 x L12+ l13 x L13 = (-1x-2)–(2x-5)+(1x1) = 13
−1 2 1
[
. L11 1
2
−1
0 2 =
1 −1
2 1
][
0 2 =−2
1 −1 ]
[
L12 1
2
−1
0 2 =
1 −1
2 1
][
1 2 =−5
2 −1 ]
[
L13 1
2
0 2 =
1 −1 ][
1 0 =1
2 1 ]
Matriz de Menores:
−1 2 1
[
M 11 1
2
−1
0
1
2
][
2 =
−1
1
0 2 =−2
1 −1 ]
M 12 1
−1
[
2
0
1
2
−1
1
][
2 =
1 2 =−5
2 −1 ]
[
M 13 1
2
−1
0
1
2
][
2 =
−1
1
1 0 =1
2 1 ]
[
M 21 1
2
−1
0
1
2
][
2 =
−1
1
2 1 =−3
1 −1 ]
−2 −5 1
M 22 1
[
2
0
1 −1][
2 =
−1 1 =−1
2 −1 ] [
M = −3 −1 −5
4 −3 −2 ]
−1 2 1
[
M 23 1
−1
2
0
1
2
2 =
−1
1
][
−1 2 =−5
2 1 ]
[
M 31 1
2
−1
0
1
2
2 =
−1
1
][
2 1 =4
0 2 ]
[
M 32 1
−1
2
0
1
2
2 =
−1
1
][
−1 1 =−3
1 2 ]
[
M 33 1
2
0
1
2 =
−1 ][
−1 2 =−2
1 0 ]
[
LT = 5 −1 3
1 5 −2 ] 1
Matriz Inversa: L =LT x ( )
−1
d
L−111 =LT 11 x ( 1d )
=−2 x
1
13 ( )
=0.153846
1 1
L−112=LT 12 x( ) =3 x( )=0.230769
d 13
L−113=LT 13 x ( 1d )=4 x( 131 )=0.307692
1 1
L−121=LT 21 x ( ) =5 x ( )=0.384615
d 13
1 1
L−122=LT 22 x ( ) =−1 x ( ) =−0.076923
d 13
1 1
L−123=LT 23 x ( )=3 x ( )=0.230769
d 13
1 1
L−131=LT 31 x ( ) =1 x ( )=0.076923
d 13
1 1
L−132=LT 32 x ( ) =5 x ( ) =0.384615
d 13
1 1
L−133=LT 33 x ( )=−2 x ( )=−0.153846
d 13
RPTA:
0.153846 0.230769 0.307692
[
L−1= 0.384615 −0.076923 0.230769
0.076923 0.384615 −0.153846 ]
1 2 −5
22. C=
[ 2 3 −8
−1 −1 5 ]
Det. (C) :c 11 x C 11−c12 x C 12+c 13 x C13 = (1x7)–(2x2)+(-5x1) = -2
1 2 −5
[
.C 11 2 3 −8 =
−1 −1 5
1 2 −5
−1 5 ][
3 −8 =7
]
[
C 12 2 3 −8 =
−1 −1 5
1 2 −5
−1 5 ][
2 −8 =2
]
[
C 13 2 3 −8 =
−1 −1 5
2
−1 −1 ][
3 =1
]
Matriz de Menores:
1 2 −5
[
M 11 2 3 −8 =
−1 −1 5
3 −8 =7
−1 5 ][ ]
1 2 −5
[
M 12 2
−1
1
3
−1
2
−8 =
5
−5
][
2 −8 =2
−1 5 ]
[
M 13 2
−1
1
3
−1
2
−8 =
5
−5
2
][3 =1
−1 −1 ]
[
M 21 2
−1
1
3
−1
2
−8 =
5
−5
][
2 −5 =5
−1 5 ]
7 2 1
[
M 22 2
−1
1
3
−1
2
−8 =
5
−5
][
1 −5 =0
−1 5 ] [
M= 5 0 1
−1 2 −1 ]
[
M 23 2
−1
1
3
−1
2
−8 =
5
−5
1
][2 =1
−1 −1 ]
[
M 31 2
−1
1
3
−1
2
−8 =
5
−5
][
2 −5 =−1
3 −8 ]
[
M 32 2
−1
1
3
−1
2
−8 =
5
−5
][
1 −5 =2
2 −8 ]
[
M 33 2
−1
3
−1
−8 =
5 ][
1 2 =−1
2 3 ]
[
C = −2 0 −2
1 −1 −1 ] 1
Matriz Inversa:C =C T x( )
−1
d
C−111=C T 11 x ( 1d ) ( )
=7 x
1
−2
=−3.5
1 1
C−112=CT 12 x( )
=−5 x ( )=2.5
d −2
1 1
C−113=CT 13 x( )
=−1 x ( ) =0.5
d −2
1 1
C−121=CT 21 x( )
=−2 x ( ) =1
d −2
1 1
C−122=CT 22 x( )
=0 x ( ) =0
d −2
1 1
C−123=CT 23 x( )
=−2 x ( )=1
d −2
1 1
C−131=CT 31 x( )
=1 x ( ) =−0.5
d −2
1 1
C−132=CT 32 x( )
=−1 x ( ) =0.5
d −2
1 1
C−133=CT 33 x( )
=−1 x ( ) =0.5
d −2
RPTA:
−3.5 2.5 0.5
C−1=
[1 0 1
−0.5 0.5 0.5 ]
−3 5 8
23. M =
[ ]
2 4 6
−1 7 4
[
M 12 2
−1
−3
4 6=
7 4
5 8
]2
−1[ 6 =14
4 ]
[
M 13 2
−1
4 6=
7 4 ][
2
−1
4 =1
7 ]
8
Matriz de Menores:
−3 5 8
[
M 11 2
−1
−3
4
7
5
4][
6 =
8
4 6 =−26
7 4]
[
M 12 2
−1
−3
4
7
5 8
][
6=
4
2 6 =14
−1 4 ]
[
M 13 2
−1
−3
4
7
5
4
8
][
6=
2 4 =18
−1 7 ]
[
M 21 2
−1
−3
4
7
5
4][
6=
8
5 8 =−36
7 4]
−26 14 18
[
M 22 2
−1
−3
7
5
4
8
][
6=
4
−3 8 =−4
−1 4 ] [
M = −36 −4 −16
−2 −34 −22 ]
[
M 23 2
−1
−3
4
7
5
4
8
][
6=
−3 5 =−16
−1 7 ]
[
M 31 2
−1
−3
4
7
5
4][
6=
8
5 8 =−2
4 6]
[
M 32 2
−1
4
7 ][
6=
4
−3 8 =−34
2 6 ]
−3 5 8
[−1 7 4 ][
M 33 2 4 6 = −3 5 =−22
2 4 ]
[
M T = −14 −4 34
18 16 −22 ] 1
Matriz Inversa: M =M T x ( )
−1
d
M −111=M T 11 x ( d1 )
=−26 x
1
152( )
=−0.171053
1 1
M −112=M T 12 x( ) =36 x ( )=0.236842
d 152
1 1
M −113=M T 13 x( ) =−2 x ( ) =−0.013158
d 152
1 1
M −121=M T 21 x( ) =−14 x ( ) =−0.092105
d 152
1 1
M −122=M T 22 x( ) =−4 x ( ) =−0.026316
d 152
1 1
M −123=M T 23 x( ) =34 x ( )=0.223684
d 152
1 1
M −131=M T 31 x( ) =18 x ( ) =0.118421
d 152
1 1
M −132=M T 32 x( ) =16 x ( ) =0.105263
d 152
1 1
M −133=M T 33 x( ) =−22 x ( ) =−0.144737
d 152
RPTA:
−0.171053 0.236842 −0.013158
−1
[
M = −0.092105 −0.026316 0.223684
0.118421 0.105263 −0.144737 ]
2 −2 1
24. J=
[ 4 −4 3
6 5 −3 ]
Det. (J) : j11 x J 11− j12 x J 12 + j13 x J 13 = (2x-3)–(-2x-30)+(1x44) =-22
2 −2 1
J
[
. 11 4
6
2
−4 3 =
5 −3
−2 1
][
−4 3 =−3
5 −3 ]
[
J 12 4
2
6
−4 3 =
5 −3
−2 1
][
4 3 =−30
6 −3 ]
[
J 13 4
6
−4 3 =
5 −3 ][
4 −4 =44
6 5 ]
Matriz de Menores:
2 −2 1
[
M 11 4
6
2
−4
5
−2
][
3 =
−3
1
−4 3 =−3
5 −3 ]
M 12 4
2
[
6
−4
5
−2
−3
1
][
3 =
4 3 =−30
6 −3 ]
[
M 13 4
6
2
−4
5
−2
][
3 =
−3
1
4 −4 =44
6 5 ]
[
M 21 4
6
2
−4
5
−2
][
3 =
−3
1
−2 1 =1
5 −3 ]
−3 −30 44
M 22 4
2
[
6
−4
5
−2
−3
1
][
3 =
2 1 =−12
6 −3 ] [
M = 1 −12 22
−2 2 0 ]
[
M 23 4
6
2
−4
5
−2
][
3 =
−3
1
2 −2 =22
6 5 ]
[
M 31 4
6
2
−4
5
−2
][
3 =
−3
1
−2 1 =−2
−4 3 ]
M 32 4
[
6
−4
5 ][
3 =
−3
2 1 =2
4 3 ]
2 −2 1
M 33
[ 4 −4 3 =
6 5 −3
4 −4 ][
2 −2 =0
]
[
J = 30 −12 −2
44 −22 0 ] 1
Matriz Inversa: J =J T x( )
−1
d
J −111 =J T 11 x ( 1d )
=−3 x
1
( )
−22
=0.136364
1 1
J −112=J T 12 x( ) =−1 x ( )=0.045455
d −22
1 1
J −113=J T 13 x( ) =−2 x ( )=0.090909
d −22
1 1
J −121=J T 21 x( ) =30 x ( )=−1.363636
d −22
1 1
J −122=J T 22 x( ) =−12 x ( )=0.545455
d −22
1 1
J −123=J T 23 x( ) =−2 x ( )=0.090909
d −22
1 1
J −131=J T 31 x( ) =44 x ( )=−2
d −22
J −132=J T 32 x ( 1d )=−22 x (−22
1
)=1
1 1
x ( )=0 x (
−22 )
J −133=J T 33 =0
d
RPTA:
0.136364 0.045455 0.090909
[
J −1 = −1.363636 0.545455 0.090909
−2 1 0 ]
1 2 −1
25. Z=
[ ] 3 4 2
1 0 −2
[ ]
. Z11 3
1
1
4
0
2
−2
−1
2 = [4
0
2 =−8
−2 ]
[ ][
Z12 3
1
1
2
4
0 −2
−1
2 =
3
1
2 =−8
−2 ]
[ ][
Z13 3
1
4
0
2 =
−2
3
1
4 =−4
0]
Matriz de Menores:
1 2 −1
[ ][
M 11 3
1
1
4
0
2
2 =
−2
−1
4 2 =−8
0 −2 ]
[ ][
M 12 3
1
1
4
0
2
2 =
−2
−1
3 2 =−8
1 −2 ]
[ ][
M 13 3
1
1
4
0
2
2 =
−2
−1
3 4 =−4
1 0 ]
[ ][
M 21 3
1
1
4
0
2
2 =
−2
−1
−2 −1 =−4
0 −2 ]
−8 −8 −4
[ ][
M 22 3
1
4
0
2 =
−2
1 −1 =−1
1 −2 ] [
M = −4 −1 −2
8 5 −2 ]
1 2 −1
[
M 23 3
1
1
4
0
2
2 =
−2
−1
]
1
1 [ 20]=−2
[
M 31 3
1
1
4
0
2
2 =
−2
−1
][
2
4
−1 =8
2 ]
M 32 3
1
1
[ 4
0
2
2 =
−2
−1
1
3][ −1 =5
2 ]
[
M 33 3
1
4
0
2 =
−2 ][
1
3
2 =−2
4]
[
Z = 8 −1 −5
−4 2 −2 ] 1
Matriz Inversa: Z =Z T x ( )
−1
d
Z−111 =Z T 11 x ( 1d )
=−8 x
1
( )
12
=−0.666667
1 1
Z−112=Z T 12 x( ) =4 x ( )=0.333333
d 12
Z−113 =Z T 13 x ( 1d )=8 x ( 121 )=0.666667
1 1
Z−121=Z T 21 x ( ) =8 x ( ) =0.666667
d 12
1 1
Z−122=Z T 22 x ( ) =−1 x ( )=−0.083333
d 12
1 1
Z−123=Z T 23 x ( )=−5 x ( )=−0.416667
d 12
1 1
Z−131=Z T 31 x ( ) =−4 x ( ) =−0.333333
d 12
1 1
Z−132=Z T 32 x ( ) =2 x ( )=0.166667
d 12
1 1
Z−133 =Z T 33 x ( )=−2 x ( ) =−0.166667
d 12
RPTA:
−0.666667 0.333333 0.666667
[
Z−1= 0.666667 −0.083333 −0.416667
−0.333333 0.166667 −0.166667 ]
27.N=
[
. N 11 3
0
1
6
−1 4
5 =
2 −5
][
6 5 =29
−1 4 ]
[
N 12 3
1
0
6
−1 4
2 −5
5 =
][
3 5 =12
0 4 ]
[
N 13 3
0
6 5 =
−1 4 ][
3 6 =−3
0 −1 ]
Matriz de Menores:
1 2 −5
[
M 11 3 6 5 =
0 −1 4
1 2 −5
−1 4 ][
6 5 =29
]
[
M 12 3 6 5 =
0 −1 4
3 5 =12
0 4 ][ ]
1 2 −5
[
M 13 3
0
1
6
−1
2
4
−5
][
5 =
3 6 =−3
0 −1 ]
[
M 21 3
0
1
6
−1
2
4
−5
][
5 =
2 −5 =3
−1 4 ]
29 12 −3
[
M 22 3
1
0
6
−1
2
][
5 =
4
−5
1 −5 = 4
0 4 ] [
M = 3 4 −1
40 20 0 ]
[
M 23 3
0
1
6
−1
2
4
−5
][
5 =
1 2 =−1
0 −1 ]
[
M 31 3
0
1
6
−1
2
4
−5
][
5 =
2 −5 = 40
6 5 ]
[
M 32 3
0
1
6
−1
2
4
−5
][
5 =
1 −5 =20
3 5 ]
[
M 33 3
0
6
−1 ][
5 =
4
1 2 =0
3 6 ]
−3[
N = −12 4 −20
1 0 ] 1
Matriz Inversa: N =N T x( )
−1
d
N −111 =N T 11 x ( 1d ) ( )
=29 x
1
20
=1.45
1 1
N −112=N T 12 x( )
=−3 x ( ) =−0.15
d 20
1 1
N −113=N T 13 x( )
=40 x ( ) =2
d 20
1 1
N −121=N T 21 x( )
=−12 x ( ) =−0.6
d 20
1 1
N −122=N T 22 x( )
=4 x ( ) =0.2
d 20
1 1
N −123=N T 23 x( )
=−20 x ( ) =−1
d 20
1 1
N −131=N T 31 x( )
=−3 x ( ) =−0.15
d 20
1 1
N −132=N T 32 x( )
=1 x ( ) =0.05
d 20
1 1
N −133=N T 33 x( )
=0 x ( ) =0
d 20
RPTA:
1.45 −0.15 2
[
N −1 = −0.6 0.2 −1
−0.15 0.05 0 ]
28.
Det. (A) :a 11 x A 11 −a12 x A 12+ a13 x A 13 = (2x0)–(0x3)+(1x3)= 3
2 0 1
[ ][
. A11 3
5
2
0
1
0
0=
1
1
0 0 =0
1 1 ]
[ ][
A12 3
5
0
1
0=
1
3 0 =3
5 1 ]
2 0 1
A13
[3 0 0=
5 1 1 ][
3 0 =3
5 1 ]
Matriz de Menores:
2 0 1
M 11 3
5
2
[ 0
1
0
][
0=
1
1
0
1
0 =0
1]
M 12 3
5
2
[ 0
1
0
][
1
0=
1
3
5
0 =3
1 ]
M 13 3
5
2
[ 0
1
0
][
0=
1
1
3
5
0 =3
1]
M 21 3
5
2
[ 0
1
0
][
0=
1
1
0
1
1 =−1
1]
0 3 3
M 22 3
5
2
[ 0
1
0
][
1
0=
1
2
5
1 =−3
1 ] [
M = −1 −3 2
0 −3 0 ]
M 23 3
5
2
[ 0
1
0
][
0=
1
1
2
5
0 =2
1]
M 31 3
5
2
[ 0
1
0
][
0=
1
1
0
0
1 =0
0]
M 32 3
5
2
[ 0
1
0
][
1
0=
1
2
3
1 =−3
0 ]
M 33 3
5 [ 0
1 ][
0=
1
2
3
0 =0
0]
[
AT = −3 −3 3
3 −2 0 ] 1
Matriz Inversa: A =A T x ( )
−1
d
A−111 = AT 11 x ( d1 )=0 x( 13 )=0
1 1
A−112= A T 12 x ( )=1 x ( )=0.333333
d 3
1 1
A−113= A T 13 x ( ) =0 x ( )=0
d 3
1 1
A−121= A T 21 x ( )=−3 x ( ) =−1
d 3
1 1
A−122= A T 22 x ( )=−3 x ( ) =−1
d 3
1 1
A−123= A T 23 x ( ) =3 x ( )=1
d 3
1 1
A−131= A T 31 x ( )=3 x ( ) =1
d 3
1 1
A−132= A T 32 x ( )=−2 x ( )=−0.666667
d 3
1 1
A−133= A T 33 x ( ) =0 x ( )=0
d 3
RPTA:
0 0.333333 0
−1
A = −1
[−1 1
1 −0.666667 0 ]
1 2 3
A 0 0 0 Toda la fila es 0 por lo
tanto Det A = 0
4 8 6
29.
Det. (A) :a 11 x A 11 −a12 x A 12+ a13 x A 13 = (1x0)–(-2x0)+(3x0)= 0
1 −2 3
. A11 0
4
1
[ 0
−8
−2
][
0=
6
3
0 0 =0
−8 6 ]
A12 0
1
4[ 0
−8
−2 3
][
0=
6
0 0 =0
4 6]
A13 0
4[ 0
−8 ][
0=
6
0 0 =¿
4 −8 ]
0
30.
[
.C 11 6 −1 1 =
−4 0
8
0
6 −6
0 0][
−1 1 =0
]
[
−4 0
8 6 −6
0
−4 0][
C 12 6 −1 1 = 6 1 =4 ]
[
−4 0 0
−4 0][
C 13 6 −1 1 = 6 −1 =−4 ]
Es imposible calcular la matriz inversa, la matriz es degenerada (el determinante de la
matriz es igual a cero)