This document provides formulas and definitions for calculus, including: - Derivatives and integrals of basic functions - Exponent rules - Logarithm rules - Trigonometric functions and their inverses - Trigonometric identities - Hyperbolic functions and their inverses - Graphs of trigonometric and hyperbolic functions

1. Formulario de Cálculo Diferencial e Integral Jesús Rubí M.
Formulario de
Cálculo Diferencial
e Integral VER.4.3
Jesús Rubí Miranda (jesusrubim@yahoo.com)
http://mx.geocities.com/estadisticapapers/
http://mx.geocities.com/dicalculus/
VALOR ABSOLUTO
1 1
1 1
si 0
si 0
y
0 y 0 0
ó
ó
n n
k k
k k
n n
k k
k k
a a
a
a a
a a
a a a a
a a a
ab a b a a
a b a b a a
= =
= =
≥⎧
= ⎨
− <⎩
= −
≤ − ≤
≥ = ⇔ =
= =
+ ≤ + ≤
∏ ∏
∑ ∑
EXPONENTES
( )
( )
/
p q p q
p
p q
q
qp pq
p p p
p p
p
qp q p
a a a
a
a
a
a a
a b a b
a a
b b
a a
+
−
⋅ =
=
=
⋅ = ⋅
⎛ ⎞
=⎜ ⎟
⎝ ⎠
=
LOGARITMOS
10
log
log log log
log log log
log log
log ln
log
log ln
log log y log ln
x
a
a a a
a a a
r
a a
b
a
b
e
N x a
MN M N
M
M N
N
N r N
N N
N
a a
N
N N N N
= ⇒
= +
= −
=
= =
= =
=
ALGUNOS PRODUCTOS
ac ad= +( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( )
( )
( )
2 2
2 2 2
2 2 2
2
2
3 3 2 2 3
3 3 2 2 3
2 2 2 2
2
2
3 3
3 3
2 2 2
a c d
a b a b a b
a b a b a b a ab b
a b a b a b a ab b
x b x d x b d x bd
ax b cx d acx ad bc x bd
a b c d ac ad bc bd
a b a a b ab b
a b a a b ab b
a b c a b c ab ac bc
⋅ +
+ ⋅ − = −
+ ⋅ + = + = + +
− ⋅ − = − = − +
+ ⋅ + = + + +
+ ⋅ + = + + +
+ ⋅ + = + + +
+ = + + +
− = − + −
+ + = + + + + +
1
1
n
n k k n n
k
a b a ab b a b
a b a a b ab b a b
a b a a b a b ab b a b
a b a b a b n− −
=
− ⋅ + + = −
− ⋅ + + + = −
− ⋅ + + + + = −
⎛ ⎞
− ⋅ = − ∀ ∈⎜ ⎟
⎝ ⎠
∑
( ) ( )
( ) ( )
( ) ( )
( )
2 2 3 3
3 2 2 3 4 4
4 3 2 2 3 4 5 5
( ) ( )
( ) ( )
( ) ( )
( ) ( )
2 2 3 3
3 2 2 3 4 4
4 3 2 2 3 4 5 5
5 4 3 2 2 3 4 5 6 6
a b a ab b a b
a b a a b ab b a b
a b a a b a b ab b a b
a b a a b a b a b ab b a b
+ ⋅ − + = +
+ ⋅ − + − = −
+ ⋅ − + − + = +
+ ⋅ − + − + − = −
( ) ( )
( ) ( )
1 1
1
1 1
1
1 impar
1 par
n
k n k k n n
k
n
k n k k n n
k
a b a b a b n
a b a b a b n
+ − −
=
+ − −
=
⎛ ⎞
+ ⋅ − = + ∀ ∈⎜ ⎟
⎝ ⎠
⎛ ⎞
+ ⋅ − = − ∀ ∈⎜ ⎟
⎝ ⎠
∑
∑
SUMAS Y PRODUCTOS
n
( )
( )
1 2
1
1
1 1
1 1 1
1 0
n k
k
n
k
n n
k k
k k
n n n
k k k k
k k k
n
k k n
k
a a a a
c nc
ca c a
a b a b
a a a a
=
=
= =
= = =
−
=
+ + + =
=
=
+ = +
− = −
∑
∑
∑ ∑
∑ ∑ ∑
∑
( )
1
( )
( )
( )
( )
( )
( )
( )
1
1
1
2
1
2 3 2
1
3 4 3 2
1
4 5 4 3
1
2
1
1 2 1
2
=
2
1
1 1
1
2
1
2 3
6
1
2
4
1
6 15 10
30
1 3 5 2 1
!
n
k
nn
k
k
n
k
n
k
n
k
n
k
n
k
n
a k d a n d
n
a l
r a rl
ar a
r r
k n n
k n n n
k n n n
k n n n n
n n
n k
n n
k
=
−
=
=
=
=
=
=
+ − = + −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
+
− −
= =
− −
= +
= + +
= + +
= + + −
+ + + + − =
=
⎛ ⎞
=⎜ ⎟
⎝ ⎠
∑
∑
∑
∑
∑
∑
∏
( )
( )
0
!
,
! !
n
n n k k
k
k n
n k k
n
x y x y
k
−
=
≤
−
⎛ ⎞
+ = ⎜ ⎟
⎝ ⎠
∑
( ) 1 2
1 2 1 2
1 2
!
! ! !
k
n nn n
k k
k
n
x x x x x x
n n n
+ + + = ⋅∑
CONSTANTES
9…3.1415926535
2.71828182846e
π =
= …
TRIGONOMETRÍA
1
sen csc
sen
1
cos sec
cos
sen 1
tg ctg
cos tg
CO
HIP
CA
HIP
CO
CA
θ θ
θ
θ θ
θ
θ
θ θ
θ θ
= =
= =
= = =
radianes=180π
CA
CO
HIP
θ
θ sen cos tg ctg sec csc
0 0 1 0 ∞ 1 ∞
30 1 2 3 2 1 3 3 2 3 2
45 1 2 1 2 1 1 2 2
60 3 2 1 2 3 1 3 2 2 3
90 1 0 ∞ 0 ∞ 1
[ ]
[ ]
sen ,
2 2
cos 0,
tg ,
2 2
1
ctg tg 0,
1
sec cos 0,
1
csc sen ,
2 2
y x y
y x y
y x y
y x y
x
y x y
x
y x y
x
π π
π
π π
π
π
π π
⎡ ⎤
= ∠ ∈ −⎢ ⎥
⎣ ⎦
= ∠ ∈
= ∠ ∈ −
= ∠ = ∠ ∈
= ∠ = ∠ ∈
⎡ ⎤
= ∠ = ∠ ∈ −⎢ ⎥
⎣ ⎦
Gráfica 1. Las funciones trigonométricas: sen x ,
cos x , tg x :
-8 -6 -4 -2 0 2 4 6 8
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
sen x
cos x
tg x
Gráfica 2. Las funciones trigonométricas csc x ,
sec x , ctg x :
-8 -6 -4 -2 0 2 4 6 8
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
csc x
sec x
ctg x
Gráfica 3. Las funciones trigonométricas inversas
arcsen x , arccos x , arctg x :
-3 -2 -1 0 1 2 3
-2
-1
0
1
2
3
4
arc sen x
arc cos x
arc tg x
Gráfica 4. Las funciones trigonométricas inversas
arcctg x , arcsec x , arccsc x :
-5 0 5
-2
-1
0
1
2
3
4
arc ctg x
arc sec x
arc csc x
IDENTIDADES TRIGONOMÉTRICAS
2
cos 1θ θ
2
2 2
2 2
sen
1 ctg csc
tg 1 sec
θ θ
θ θ
+ =
+ =
+ =
( )
( )
( )
sen sen
cos cos
tg tg
θ θ
θ θ
θ θ
− = −
− =
− = −
( )
( )
( )
( )
( )
( )
( ) ( )
( ) ( )
( )
sen 2 sen
cos 2 cos
tg 2 tg
sen sen
cos cos
tg tg
sen 1 sen
cos 1 cos
tg tg
n
n
n
n
n
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
+ =
+ =
+ =
+ = −
+ = −
+ =
+ = −
+ = −
+ =
( )
( ) ( )
( )
( )
sen 0
cos 1
tg 0
2 1
sen 1
2
2 1
cos 0
2
2 1
tg
2
n
n
n
n
n
n
n
n
π
π
π
π
π
π
=
= −
=
+⎛ ⎞
= −⎜ ⎟
⎝ ⎠
+⎛ ⎞
=⎜ ⎟
⎝ ⎠
+⎛ ⎞
= ∞⎜ ⎟
⎝ ⎠
sen cos
2
cos sen
2
π
θ θ
π
θ θ
⎛ ⎞
= −⎜ ⎟
⎝ ⎠
⎛ ⎞
= +⎜ ⎟
⎝ ⎠
( )
( )
( )
( )
( )
2 2
2
2
2
2
sen sen cos cos sen
cos cos cos sen sen
tg tg
tg
1 tg tg
sen 2 2sen cos
cos2 cos sen
2tg
tg 2
1 tg
1
sen 1 cos2
2
1
cos 1 cos2
2
1 cos2
tg
1 cos2
α β α β α β
α β α β α β
α β
α β
α β
θ θ θ
θ θ θ
θ
θ
θ
θ θ
θ θ
θ
θ
θ
± = ±
± =
±
± =
=
= −
=
−
= −
= +
−
=
+
∓
∓
( ) ( )
( ) ( )
( ) ( )
( ) ( )
1 1
sen sen 2sen cos
2 2
1 1
sen sen 2sen cos
2 2
1 1
cos cos 2cos cos
2 2
1 1
cos cos 2sen sen
2 2
α β α β α β
α β α β α β
α β α β α β
α β α β α β
+ = + ⋅ −
− = − ⋅ +
+ = + ⋅ −
− = − + ⋅ −
( )sen
tg tg
cos cos
α β
α β
α β
±
± =
⋅
( ) ( )
( ) ( )
( ) ( )
1
sen cos sen sen
2
1
sen sen cos cos
2
1
cos cos cos cos
2
α β α β α β
α β α β α β
α β α β α β
⋅ = − + +⎡ ⎤⎣ ⎦
⋅ = − − +⎡ ⎤⎣ ⎦
⋅ = − + +⎡ ⎤⎣ ⎦
tg tg
tg tg
ctg ctg
α β
α β
α β
+
⋅ =
+
FUNCIONES HIPERBÓLICAS
senh
2
cosh
2
senh
tgh
cosh
1
ctgh
tgh
1 2
sech
cosh
1 2
csch
senh
x x
x x
x
x
x x
x x
x x
x
e e
x
x e e
x
x e e
e e
x
x e e
x
x x
e e
x
x
x e e
x
x e e
−
−
−
−
−
−
−
=
+
=
−
= =
+
+
= =
−
= =
+
= =
−
−
−
[
{ }
]
{ } { }
senh :
cosh : 1,
tgh : 1,1
ctgh : 0 , 1 1,
sech : 0,1
csch : 0 0
→
→ ∞
→ −
− → −∞ − ∪ ∞
→
− → −
Gráfica 5. Las funciones hiperbólicas senh x ,
cosh x , tgh x :
-5 0 5
-4
-3
-2
-1
0
1
2
3
4
5
senh x
cosh x
tgh x
FUNCIONES HIPERBÓLICAS INV
( )
( )
1 2
1 2
1
1
2
1
2
1
senh ln 1 ,
cosh ln 1 , 1
1 1
tgh ln , 1
2 1
1 1
ctgh ln , 1
2 1
1 1
sech ln , 0 1
1 1
csch ln , 0
x x x x
x x x x
x
x x
x
x
x x
x
x
x x
x
x
x x
x x
−
−
−
−
−
−
= + + ∀ ∈
= ± − ≥
+⎛ ⎞
= <⎜ ⎟
−⎝ ⎠
+⎛ ⎞
= >⎜ ⎟
−⎝ ⎠
⎛ ⎞± −
⎜ ⎟= < ≤
⎜ ⎟
⎝ ⎠
⎛ ⎞+
⎜ ⎟= + ≠
⎜ ⎟
⎝ ⎠

2. Formulario de Cálculo Diferencial e Integral Jesús Rubí M.
IDENTIDADES DE FUNCS HIP
2 2
senh 1x x
( )
( )
( )
2 2
2
cosh
1 tgh sech
ctgh 1 csch
senh senh
cosh cosh
tgh tgh
x x
x x
x x
x x
x x
− =
− =
− = −
− =
− = −
− =
( )
( )
( )
2 2
2
senh senh cosh cosh senh
cosh cosh cosh senh senh
tgh tgh
tgh
1 tgh tgh
senh 2 2senh cosh
cosh 2 cosh senh
2tgh
tgh 2
1 tgh
x y x y x y
x y x y x y
x y
x y
x y
x x x
x x x
x
x
x
± = ±
± = ±
±
± =
±
=
= +
=
+
( )
( )
2
2
2
1
senh cosh 2 1
2
1
cosh cosh 2 1
2
cosh 2 1
tgh
cosh 2 1
x x
x x
x
x
x
= −
= +
−
=
+
senh 2
tgh
cosh 2 1
x
x
x
=
+
cosh senh
cosh senh
x
x
e x x
e x x−
= +
= −
OTRAS
( ) ( )
2
2
2
0
4
2
4 discriminante
exp cos sen si ,
ax bx c
b b ac
x
a
b ac
i e iα
α β β β α β
+ + =
− ± −
⇒ =
− =
± = ± ∈
LÍMITES
( )
1
0
0
0
0
1
lim 1 2.71828...
1
lim 1
sen
lim 1
1 cos
lim 0
1
lim 1
1
lim 1
ln
x
x
x
x
x
x
x
x
x
x e
e
x
x
x
x
x
e
x
x
x
→
→∞
→
→
→
→
+ = =
⎛ ⎞
+ =⎜ ⎟
⎝ ⎠
=
−
=
−
=
−
=
DERIVADAS
( )
( ) ( )
( )
( )
( )
( )
( )
0 0
1
lim lim
0
x
x x
n n
f x x f xdf y
D f x
dx x x
d
c
dx
d
cx c
dx
d
cx ncx
dx
d du dv dw
u v w
dx dx dx dx
d du
cu c
dx dx
∆ → ∆ →
−
+ ∆ − ∆
= = =
∆ ∆
=
=
=
± ± ± = ± ± ±
=
( )
( )
( ) ( )
( )
2
1n n
d dv du
uv u v
dx dx dx
d dw dv du
uvw uv uw vw
dx dx dx dx
v du dx u dv dxd u
dx v v
d du
u nu
dx dx
−
= +
= + +
−⎛ ⎞
=⎜ ⎟
⎝ ⎠
=
( )
( )
( )
( )
12
1 2
(Regla de la Cadena)
1
donde
dF dF du
dx du dx
du
dx dx du
dF dudF
dx dx du
x f tf tdy dtdy
dx dx dt f t y f t
= ⋅
=
=
=⎧′ ⎪
= = ⎨
′ =⎪⎩
DERIVADA DE FUNCS LOG & EXP
( )
( )
( )
( )
( )
( ) 1
1
ln
log
log
log
log 0, 1
ln
ln
a
a
u u
u u
v v v
d du dx du
u
dx u u dx
d e du
u
dx u dx
ed du
u a
dx u dx
d du
e e
dx dx
d du
a a a
dx dx
d du dv
u vu u u
dx dx dx
−
= = ⋅
= ⋅
= ⋅ >
= ⋅
= ⋅
= + ⋅ ⋅
a ≠
DERIVADA DE FUNCIONES TRIGO
( )
( )
( )
( )
( )
( )
( )
2
2
sen cos
cos sen
tg sec
ctg csc
sec sec tg
csc csc ctg
vers sen
d du
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u u
dx dx
d du
u u u
dx dx
d du
u u
dx dx
=
= −
=
= −
=
= −
=
DERIV DE FUNCS TRIGO INVER
( )
( )
( )
( )
( )
( )
( )
2
2
2
2
2
2
2
1
sen
1
1
cos
1
1
tg
1
1
ctg
1
si 11
sec
si 11
si 11
csc
si 11
1
vers
2
d du
u
dx dxu
d du
u
dx dxu
d du
u
dx u dx
d du
u
dx u dx
ud du
u
udx dxu u
ud du
u
udx dxu u
d du
u
dx dxu u
∠ = ⋅
−
∠ = − ⋅
−
∠ = ⋅
+
∠ = − ⋅
+
+ >⎧
∠ = ± ⋅ ⎨
− < −− ⎩
− >⎧
∠ = ⋅ ⎨
+ < −− ⎩
∠ = ⋅
−
∓
DERIVADA DE FUNCS HIPERBÓLICAS
2
2
senh cosh
cosh senh
tgh sech
ctgh csch
sech sech tgh
csch csch ctgh
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u u
dx dx
d du
u u u
dx dx
=
=
=
= −
= −
= −
d du
DERIVADA DE FUNCS HIP INV
1
2
-1
1
-12
1
2
1
2
1
1
12
senh
1
si cosh 01
cosh , 1
si cosh 01
1
tgh , 1
1
1
ctgh , 1
1
si sech 0, 0,11
sech
si sech 0, 0,11
u
dx dxu
ud du
u u
dx dx uu
d du
u u
dx u dx
d du
u u
dx u dx
u ud du
u
dx dx u uu u
−
−
−
−
−
−
−
= ⋅
+
⎧+ >± ⎪
= ⋅ > ⎨
− <− ⎪⎩
= ⋅ <
−
= ⋅ >
−
⎧− > ∈⎪
= ⋅ ⎨
+ < ∈− ⎩
∓
1d du
1
2
1
csch , 0
1
d du
u u
dx dxu u
−
⎪
= − ⋅ ≠
+
INTEGRALES DEFINIDAS, PROPIEDADES
{ }( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )
( ) [ ]
( ) ( )
( ) ( ) [ ]
( ) ( )
0
, , ,
,
si
a a a
b b
a a
b c b
a a c
b a
a b
a
a
b
a
b b
a a
b b
a a
b b b
f x g x dx f x dx g x dx
cf x dx c f x dx c
f x dx f x dx f x dx
f x dx f x dx
f x dx
m b a f x dx M b a
m f x M x a b m M
f x dx g x dx
f x g x x a b
f x dx f x dx a b
± = ±
= ⋅ ∈
= +
= −
=
⋅ − ≤ ≤ ⋅ −
⇔ ≤ ≤ ∀ ∈ ∈
≤
⇔ ≤ ∀ ∈
≤ <
∫ ∫ ∫
∫ ∫
∫ ∫ ∫
∫ ∫
∫
∫
∫ ∫
∫ ∫
INTEGRALES
( ) ( )
( )
( )
1
Integración por partes
1
1
ln
n
n
adx ax
af x dx a f x dx
u v w dx udx vdx wdx
udv uv vdu
u
u du n
n
du
u
u
+
=
=
± ± ± = ± ± ±
= −
= ≠ −
+
=
∫
∫ ∫
∫ ∫ ∫ ∫
∫ ∫
∫
∫
INTEGRALES DE FUNCS LOG & EXP
( )
( )
( ) ( )
( )
( )
2
2
0
1ln
1
ln ln
1
ln ln ln 1
1
log ln ln 1
ln ln
log 2log 1
4
ln 2ln 1
4
u u
u
u
u
u
u u
a
a a
e du e
aa
a du
aa
a
ua du u
a a
ue du e u
udu u u u u u
u
udu u u u u
a a
u
u udu u
u
u udu u
=
>⎧
= ⎨
≠⎩
⎛ ⎞
= ⋅ −⎜ ⎟
⎝ ⎠
= −
= − = −
= − = −
= ⋅ −
= −
∫
∫
∫
∫
∫
∫
∫
∫
INTEGRALES DE FUNCS TRIGO
u u
2
2
sen cos
cos sen
sec tg
csc ctg
sec tg sec
csc ctg csc
ud
udu u
udu u
udu u
u udu u
u udu u
= −
=
=
= −
=
= −
∫
∫
∫
∫
∫
∫
tg ln cos ln sec
ctg ln sen
sec ln sec tg
csc ln csc ctg
udu u u
udu u
udu u u
udu u u
= − =
=
= +
= −
∫
∫
∫
∫
( )
2
2
2
2
1
sen sen 2
2 4
1
cos sen 2
2 4
tg tg
ctg ctg
u
udu u
u
udu u
udu u u
udu u u
= −
= +
= −
= − +
∫
∫
∫
∫
sen sen cos
cos cos sen
u udu u u u
u udu u u u
= −
= +
∫
∫
INTEGRALES DE FUNCS TRIGO INV
( )
( )
2
2
2
2
2
2
sen sen 1
cos cos 1
tg tg ln 1
ctg ctg ln 1
sec sec ln 1
sec cosh
csc csc ln 1
csc cosh
udu u u u
udu u u u
udu u u u
udu u u u
udu u u u u
u u u
udu u u u u
u u u
∠ = ∠ + −
∠ = ∠ − −
∠ = ∠ − +
∠ = ∠ + +
∠ = ∠ − + −
= ∠ − ∠
∠ = ∠ + + −
= ∠ + ∠
∫
∫
∫
∫
∫
∫
INTEGRALES DE FUNCS HIP
coshu=
2
2
senh
cosh senh
sech tgh
csch ctgh
sech tgh sech
csch ctgh csch
udu
udu u
udu u
udu u
u udu u
u udu u
=
=
= −
= −
= −
∫
∫
∫
∫
∫
∫
( )
( )1
tgh lncosh
ctgh ln senh
sech tg senh
csch ctgh cosh
1
ln tgh
2
udu u
udu u
udu u
udu u
u
−
=
=
= ∠
= −
=
∫
∫
∫
∫
INTEGRALES DE FRAC
( )
( )
2 2
2 2
2 2
2 2
2 2
tg
1
ctg
1
ln
2
1
ln
2
du
u a a a
u
a a
du u a
u a
u a a u a
du a u
u a
a u a a u
= ∠
+
= − ∠
−
= >
− +
+
= <
− −
∫
∫
∫
1 u
INTEGRALES CON
( )
( )
2 2
2 2
2 2
2 2 2 2
2 2
2
2 2 2 2
2
2 2 2 2 2 2
sen
cos
ln
1
ln
1
cos
1
sec
sen
2 2
ln
2 2
du u
aa u
u
a
du
u u a
u a
du u
au a u a a u
du a
a uu u a
u
a a
u a u
a u du a u
a
u a
u a du u a u u a
= ∠
−
= −∠
= + ±
±
=
± + ±
= ∠
−
= ∠
− = − + ∠
± = ± ± + ±
∫
∫
∫
∫
∫
∫
MÁS INTEGRALES
( )
( )
2 2
2 2
sen cos
sen
cos sen
cos
au
au
au
e a bu b bu
e bu du
a b
e a bu b bu
e bu du
a b
−
=
+
+
=
+
∫
∫
au
ALGUNAS SERIES
( ) ( ) ( )( )
( )( )
( )
( )( )
( ) ( ) ( )
( )
( )
( )
( )
( )
2
0 0
0 0 0
0 0
2
2 3
3 5 7 2 1
1
2 4 6
''
'
2!
: Taylor
!
'' 0
0 ' 0
2!
0
: Maclaurin
!
1
2! 3! !
sen 1
3! 5! 7! 2 1 !
cos 1
2! 4!
nn
n n
n
x
n
n
f x x x
f x f x f x x x
f x x x
n
f x
f x f f x
f x
n
x x x
e x
n
x x x x
x x
n
x x x
x
−
−
−
= + − +
−
+ +
= + +
+ +
= + + + + + +
= − + − + + −
−
= − + − ( )
( )
( ) ( )
( )
2 2
1
2 3 4
1
3 5 7 2 1
1
1
6! 2 2 !
ln 1 1
2 3 4
tg 1
3 5 7 2 1
n
n
n
n
n
n
x
n
x x x x
x x
n
x x x x
x x
n
−
−
−
−
−
+ + −
−
+ = − + − + + −
∠ = − + − + + −
−

3. Formulario de Cálculo Diferencial e Integral Jesús Rubí M.
ÁLGEBRA LINEAL
Def. El determinante de una matriz
11 12
21 22
a a
A
a a
⎡ ⎤
= ⎢ ⎥
⎣ ⎦
está dado por
11 12
11 22 12 21
21 22
det .
a a
A a a a a
a a
= = −
Def. El determinante de una matriz
11 12 13
21 22 23
31 32 33
a a a
A a a a
a a a
⎡ ⎤
⎢ ⎥
= ⎢ ⎥
⎢ ⎥⎣ ⎦
está dado por
11 22 33 12 23 3111 12 13
13 21 32 11 23 3221 22 23
31 32 33 12 21 33 13 22 31
det .
a a a a a aa a a
a a a a a aA a a a
a a a a a a a a a
⋅ ⋅ + ⋅ ⋅
+ ⋅ ⋅ − ⋅ ⋅= =
− ⋅ ⋅ − ⋅ ⋅