1
!
"
#
$
!
% &
' ()
!
% + ,
-
$
*(
*
.
&
/ 0 / /1 /#
0 / 0 /
/ # $
0 /
/ #
0 /
/ 2
!
/ 2
! −
/
3 2
2 0 / 3 −3
%/ 3 1
0 /
& / −
3 3 − = 2 2
/ −
3 3 + = 2 2
/ −
1 (− 2) = 2
/ 7
5 3 − 7 2
−
1 0 / 3
4 = 9
/ (−5)1 + 5 = / (−3)
5 1 − (−3) + (−3)(4) = 9 2
Se pueden efectuar
como
para solucionar
4 15 65 2 ,728 9 : 5 86
2
que puede ser
0 /
%
"
!
;
" ; "
$
5
" "
<
"
M=
< m× n =
m11
m12 . .
.
m1n
m21
m22
.
.
m2 n .
.
.
.
.
.
.
.
. .mij
mm1
mm 2
m
.
.
#
m ij
M m ×n
. mmn
i − esima j − esima
m ×n
n
3
−1 − 2 2 1 R= 3 −3 4 −1
3 4 0 −3
> %3 4×3
2 1 4
R32
(−3)
R42
H=
4
2 0 80 − 30 40 − 50 − 10 H
<
2× 3
H
+ %3
+
3
&
a12 = 80 a23 = −10
P = [ Pij ]
Q = [Qij ]
Py Q
j
i Pij = Qij
2 −5 − 27 0
4
2× 2
1 3 + 2 2 = (− 3)3
7 − −
5 7
3 3 + 4 4
2×2
;
#
5 + 3z − 6z − 2
2 +z 3 4z
6
=
−4
2×2
1 4 3
&
2 + z =1 3 4 4z = 3
5 + 3z = 6 − 6 z − 2 = −4
2×2
$
1 &
5 + 3z = 6
2 + z =1 3
3z = 1 z=
1 3
z = 1− z=
− 6 z − 2 = −4 2 3
− 6 z = −2
1 3
z=
1 3
4 3 4 z= 12 1 z= 3
4z =
x, y, w, z
,
5x + 4 2 7 + 2x 6w = y − 7 3z 12 15 − 2 z &
5x + 4 = 7 + 2 x 5x − 2 x = 7 − 4 x =1
2 = 6w 2 =w 6 1 =w 3
y − 7 = 12
3z = 15 − 2 z
y = 12 + 7 3z + 2 z = 15 y = 19
z=3 &
1 x = 1, w = , y = 19, z = 3 3
1 3 + 2 2 −5+ w 3 z 4
2 − 3x − C = 7 y +1
(− 2)3
8 !
H=
1
6 −4 −8 −3
5
" #
$%&
&(
C32
&)
H 2×3 $ & 3 (− 2) $ &
5 7 & H 31 = C31 $ &
&
$'&
C 21 = H 21 $ &
y=
H =C
x, y, w, z
* + , -
!
. -
!
/ -
!
4×3 !
2×3
-01'1# #
(02
(
!
3
!
4 !
4
4
4 4 3
3 !
6 3
3 ( 5
! ! ) ,. 7-
4
8
3 !
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2
6
&
!
n
&
"
"
! n
n×n
M=
m11
m12 . .
.
m1n
m21
m22
.
.
m2 n .
.
.
.
.
.
.
.
. .mij
mn1
mn 2
.
m11 , m22 , m33 , !
.
n×n
. mnn
n×n
mnn
=
mij
i= j
&
i≠ j !
mij
m×n
&
m×n !
n×n
&;
mij
0'/
i≠ j i= j
n !
P T = [ Pij ]
m×n
P = [ Pij ]
&
n×m
i − ésima
i − ésima
P
2
7
P
8
F = 1 −2 9 3 5 4
G= 3×3
PT
5 0 0
20 19 70 12 13 80
2×3
H= 0 3 0 0 0 7
3×3
7
I4 =
1 0 0 0 0 1 0 0
J= 0 0 0 0
0 0 1 0 0 0 0 1
F
0 0
4×4
3×2
3× 3 H
?
3× 3 4× 4 J
I4
3× 2
20 12 G = 19 13 70 80 T
G 3×2
3 !
3 4
!
u = [1 , − 3]1×2 v = [7 , 5]1×2 p=
2 5
q= 2×1
4 3
2×1
r = [2, 5,−7] s = [7, − 2, 8] t = [9, 6, 5]
6 !
u + v = [1 + 7,−3 + 5] = [8 , 2]
p+q =
2+4 6 = 5+3 8
r + s + t = [2 + 7 + 9, 5 − 2 + 6,−7 + 8 + 5] = [18, 9, 6]
8
= #
!
M , N, S
M=
m11
m12 . .
.
m1n
n11
n12 . .
.
n1n
m21
m22
.
.
m2 n .
n21
n22
.
.
n2 n .
.
.
.
.
.
.
.
.
.
.
.
.
. .mij
.
.
. .nij
.
mm1
mm 2
nm1
nm 2
.
N=
. mmn
.
m ×n
.
nmn
.
m11 + n11
m12 + n12
.
.
m1n + n1n
m21 + n21
m22 + n22
.
.
m2 n + n 2 n .
.
.
.
.
.
.
.
. .mij + nij
mm1 + nm1
mm 2 + nm 2
M +N =S =
.
. mmn + nmn
.
m ×n
m ×n
3 -
3
+ 9
v = [−3, 5, 8] !
(− 2.5) (− 2.5v ) = [7.5, − 12.5,−20] "
! k
kM M
k
m×n
M = [mij ]m×n
"
kM
m×n
"
"
M
k
9
#
$
H=
&
(−3) H = (− 3)
−2 4 9 −1 3 5
2 1 4 3
−2 4 9 6 − 12 − 27 = −1 3 5 3 − 9 − 15
3 3 3 −2 4 9 − H= − = 3 2 2 −1 3 5 2 5I = 5
I=
−6 −
9 2
27 2 15 − 2
−
2 1 10 5 = 4 3 20 15
#
:
u = [u1 , u 2 , u3 ]1×3 ! v = [v1 , v2 , v3 ]1×3 u•v k u
v
u • v = [u1 , u 2 , u3 ] • [v1 , v2 , v3 ] = u1 ⋅ v1 + u 2 ⋅ v2 + u3 ⋅ v3 = k 6
2
−1
5 3
• −2 −3 3×1
5
[3, 5,−4] • [−2,7,−1] = −6 + 35 + 4 = 33
= −2 − 10 − 9 = −21 3×1
[−2,11]1×2 •
#
−3 −5
2×1
" −2 11
10
• 2×1
−3 −5
= 6 − 55 = −49 2×1
" m×n
P = [ Pij ] P ⋅Q = R
n×r
Q = [Qij ]
n
"
n
"
P
m×r
R = [ Rij ]
Q
P ⋅Q
" "
P
"
Q
−4 #
$
u = [−1, 7, 3] v = [−3, 6,−1] w = − 3
&
−5 L=
N=
2
−1 4 −3 5
2× 2
−1 6
M = 8 −3 4 5 −5 7
−3 2 −4 −1 7 − 5
2×3
3×3
2 −5
1 − 3 11 1
Ñ = 1 −3 0 −2
O = 2 − 2 13 8 3 − 4 10 7 3×2
3×4
u • v = (− 1)(− 3) + 7 ⋅ 6 + 3(− 1) = 3 + 42 − 3 = 42 L⋅N =
4
−1 4 −3 5
−3 2 −4 2×2
=
−1 7 − 5
2×3
− 1 26 − 16 4
29 − 13
2×3
(− 1)(− 3) + 4(− 1) = −1 @ (− 1)2 + 4 ⋅ 7 = 26 @ (− 1)(− 4) + 4(− 5) = −16 (− 3)(− 3) + 5(− 1) = 4 N ⋅L O⋅M
@ (− 3)2 + 5 ⋅ 7 = 29 @ (− 3)(− 4) + 5(− 5) = −13 N O
3
>
L
+
M
3
11
4
&
−1 6
1 − 3 11 1
18 − 28 69 36
M ⋅O = 8 − 3 4 5 −5 7
2 − 2 13 8 3 − 4 10 7
= 14 − 34 89 12 16 − 33 60 14
2
3×3
2 ⋅1 + (− 1)2 + 6 ⋅ 3 = 18
@
2 ⋅11 + (− 1)13 + 6 ⋅10 = 69
@
8 ⋅1 + (− 3)2 + 4.3 = 14
@
8 ⋅11 + (− 3)13 + 4 ⋅10 = 89
@
5 ⋅1 + (− 5)2 + 7 ⋅ 3 = 16
@
5 ⋅11 + (− 5)13 + 7 ⋅10 = 60
@
L⋅M
3×4
2(− 3) + (− 1)(− 2) + 6(− 4) = −28 @ 2 ⋅1 + (− 1)8 + 6 ⋅ 7 = 36 8(− 3) + (− 3)(− 2) + 4(− 4) = −34 8 ⋅1 + (− 3)8 + 4 ⋅ 7 = 12 5(− 3) + (− 5)(− 2) + 7(− 4) = −33 5 ⋅1 + (− 5)8 + 7 ⋅ 7 = 14
M ⋅L
3LN − 2 Ñ T = 3
:
3×4
−1 4 − 3 2 − 4 2 1 0 = −2 −5 −3 −2 − 3 5 −1 7 − 5
− 3 12 − 3 2 − 4 4 2 0 − 3 78 − 48 4 2 0 − = − = − 9 15 − 1 7 − 5 − 10 − 6 − 4 12 87 − 39 − 10 − 6 − 4 − 7 76 − 48 22 93 − 35
#
−1
7
5
d = [1, 3,−5] e = [4,−7, 3] f = − 2 K = 2 − 1
−7 5 −7 2 −2 1 G= H = 7 −4 3 −3 5 8 −1 5
12
& − 3e
" 2
& 5d
* 2
& 3 I T + 2k
, 2
& I ⋅ J & 7J ⋅ I
&
&d ⋅ K
2 6 −1 I= 8 7 −2 1 f 2
& 7 d − 4e & & I⋅f
&K ⋅H
1
4
−1 0 7 3 J = −3 2 4 8 −2 4 0 4 3 K 2
&H ⋅J
& 3G ⋅ I − 2 K T
$ !
5 > <
2× 2
3× 3
<
;
! <
!
9
A
!
,
&
a11 x1 + a12 x2 + ....... + a1n xn = b1 a21 x1 + a22 x2 + ....... + a2 n xn = b2 . . . am1 x1 + am 2 x2 + ....... + amn xn = bm <
m×n
2
2B
a11 a12 ... a1n a21 a22 ... a2 n . . . . . . . . . am1 am 2 ... amn &
13
x1 x2 . X = . . xn
b1 b2 . B= . . bm
2 ) a11
a12
... a1n b1
a21 .
a22 .
. .
. .
... a2 n b2 . . . . . .
am1 am 2 ... amn bm
2 % &
1
'
"
a 1 a
1 a =1 a
+
a11
3 ? "
' !
0
/
! 0
1
"
a
(− a ) a + (− a ) = 0
&
/ ;
14
a22 0
>
C
/
"
!
'
D
1
!
09
A
/
w + 3m + 2n = 3 3w − 2 m − n = 1 )
4 w − m + 3n = −4
a21 y a31 1
&
a11
'0
3
2 3
3 − 2 −1 1 4 −1 3 − 4
/
0 F2 → −3F1 + F2 / 0 F3 → −4 F1 + F3 / 1
3
0 − 11 0 − 13
2
3
−7 −8 − 5 − 16
& 3 E 30'/ B F @ E+ E 303/ B E ''@ E ' E 30+/
B E G @ 'E 303/ B E H > E >0'/ B F @ E ' E >03/ B E '3@ 3E >0+/ B E C @ E >E >03/ B E 'D E ''
'0
E ''
+/
15
1
3
2
3
0 F1 → −3F2 + F1 /
7 8 0 1 11 11 0 − 13 − 5 − 16
1 0
9 11 8 11 72 11
1 11 7 11 36 11
−
3 − 3(118 ) =
9 11
0 1 0 0
− 16 + (13)(118 ) =
& ' E 30F/ B '@ 3 E 30'/ B F @ 2 − (3)(117 ) =
1 11
@
F I '3 0F/ B F @ E '3 I '30'/ B F @ − 5 + (13)(117 ) =
36 11
@
−72 11
3DJ''
1 0 0 1 0 0
1 11 7 11
'0
9 11 8 11
0 F1 →
−1 11
''J3D
F3 + F1 /
0 F2 →
1 11
−1 + (11 )(1) = 0 @
−1 + (11 )(− 2) = 1
& 0 + ( −117 )(0) = 0 @ 1 + ( −117 )(0) = 1 @
7 11
& K B '@
1
!
+ ( −117 )(1) = 0 @ 118 + ( −117 )(− 2) = 2 B +@
09
B E+
A
/
5 − 11 43 4 22
5m − 11n = 43 4m + 2n = 2 5'
?
16
F3 + F2 /
−1 & 1 + ( 11 )(0) = 1 @ 0 + (11−1 )(0) = 0 @
0 1 0 2 0 0 1 −2
:
−7 11
3/
1 −2
1 0 0 1
9 11
0 F3 → 13F2 + F3 /
1
−11 43 5 5
4
22
L
F2 → −4 F1 + F2
1 5
1
−11 5
0
54 5
43 5
−
1(− 4) + 4 = 0 @ (− 4)( −511 ) + 105 =
162 5
@ (− 4) 435 + 105 = − 162 5
5 54
? 1 0 11 5
54 5
−11 5
43 5
F1 → 115 F2 + F1
1 −3
⋅ 0 +1 = 1 @
11 5
⋅1 − 115 = 0 @
11 5
1 0 2 0 1−3
(− 3) + 435 = 2 m=2
y n = −3
9 ;
<
:
(
"
10 x + 10 y = 100
&
10 x − 10 y = 80 30 x + 30 y = 50
&
20 x + 20 y = 100
2 x − 6 y = 40
&
4 x + 2 y = 50
* ;
<
:
2x − 3 y − 2z = 2 & x − y − 5 z = 18
x+ y−z = 6 m − n − 3z = −4 & 2 m − n − 4 z = −7
m + n − z = −2
17
3
=
#
!
!
−2 u = [−2,−5] v = [3,−5,−1] w = − 6
4 −1 1 C= 0 2
A=
1 −2 0 3
B=
−2 3 0 1 −4 1
1 0 0
3 D= 0 1 1 4 1 2 1
" -
9 & − 3u
& 3v
& 5v • w
&u ⋅ A &v⋅ D
* 2
: & 3 A − 2 BC & BT B & B (D + E ) & D 2
,
7 m + 2n = 8 3m − 5n = 21 .
;
2w + 3z = 17 3w + 4 z = 24
" &
18
><
:
&
& <
;
&
?
"
& "
#
&
#
& "
"
. "
! 9
4
A
. ?
2
-
#
) 4
:
+FF'
?
L
=4 : ? # : 5 'MMM ) L
,
N
O
6
2
P
'MM+
19
,4
9
O
)
) .
? 9 1
5 <
?
!
): O
2 K4 2
+FF' .
+FF+
) P
.
Q
O
Q?
2 ?
9
K 4
=
#
$ $
20
'MMH
$
5
> 5 C ,
) 5
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
2
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR .
)
RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
& RRRRRRRRRRRRRRRRR & RRRRRRRRRRRRRRR
' :
&
/; / /, /, /,
0 / 0 / 0 / 0 / 0 /
,
&
1 u = [−7,−2] v = [9,−5, 6] w = [−8,7,−1] z = 3
5 −3 −2 A= 0 −5
−1 1 1 2 0 −2 0 3 B= C= 0 3 D= 0 1 1 −1 − 4 1 2 4 1 0 1
3 0 0
1 3
0 0
1 0 0
E = 0 6 0 F = 0 16 0 I 3 = 0 1 0 0 0 3 0 0 13 0 0 1 v•w
+ 2 H
O
2H
O 'M
v• z
3
>: C 4
&
''3
)
'FC
, '+'
& ) ''3
, +>
& 4D − 2E & / 3 A − 2 BC
/ DI − 13 E / D 2
/ B (D + E )
/ EF
21
"
*
* 2 O ) ,
D #
5 x − 6 y = 13 7 x + 3 y = −16
9
'
+
A
3
2
3
O
x
)
,
y
G #
12m + 3n = 33 3m + 14n = 48
9
'
+
2
3 O
m
A
3 )
,
n
H ) !
!
9
M 1
A
!
9
A
2 x − 3 y + z = −2 3x + 3 y − 2 z = 2 x − 6 y + 3 z = −2 'F ) !
22
!
9
A
&