10 Matriz Adjunta
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Bertha Georgina Valle Cueva N. 540807
Sea A una matriz de orden n y sea B la matriz formada por los cofactores de A. Entonces la adjunta de A, que se denota adjunta A, escrito adj A, es la matriz transpuesta de la matriz B
Sea A =
2 4 3
4 6 5 6 1 -2
obtener adjunta A
1.- Primero se calculan los cofactores de cada miembro de la matriz. Cof A11 = (-1)1+15 1
6 2
= ( 5)(-2) – (1)(6) = -16
Sea A =
2 4 3
4 6 5 6 1 -2
Cof A12 = (-1)1+24 26 3
6 2
= - ( 4)(-2)- (3)(6) =
Sea A =
2 4 3
4 6 5 6 1 -2
Cof A13 = (-1)1+34 -11 3
5 1
= ( 4)(1) – (3)(5) =
Cof A21 = (-1)2+1 4 1
6 2
= -( ( 4)(-2) – (1)(6)) = 14
Cof A22 = (-1)2+2 2 3
6 2
= ( 2)(-2) – (3)(6) = -22
Cof A23 = (-1)2+3 2 3
4 1
= -( ( 2)(1) – (3)(4)) = 10
Cof A31 = (-1)3+1 4 5
6 6
= ( 4)(6) – (5)(6) = -6
Cof A32 = (-1)3+2 2 4
6 6
= -( ( 2)(6) – (4)(6)) = 12
Cof A11 = (-1)3+3 2 4
4 5
= ( 2)(5) – (4)(4) = -6
2.- Se crea la matriz B con los cofactores que se obtuvieron de la matriz A
B
=
-16
26
-11
14
-22
10
-6
12
-6
Bt
3.- Se crea la matriz B transpuesta.
=
-16
14
-6
26
-22
12
-11
-6
-6
ADJUNTA A
Sea A =
Cof A11 = =0
1
-3
0
2
3
-2
-6
-2
-1 2 10
2
5
-1
6
1
3
obtener adjunta A
-12
-2
-6
(-1)210
2
5
1
3
6
Cof A12 = -1
Cof A13 = = 0
3
-2
-6
-2
2
5
-1
1
3
(-1)
3
3
-12
-6
(-1)4-2
10
5
6
3
-1
=
Cof A14 = = 2
3
-12
-2
(-1)5-2
10
2
6
1
-1
Cof A21 = -1 Cof A22 = 1
Cof A31 =
0
Cof A41 = -2
Cof A32 =
2
Cof A42 = -2
Cof A23 = -1
Cof A33 = -3
Cof A43 =
Cof A24 =
Cof A34 = -3
Cof A44 = 2
-2
3
0
-1
0
2
-1
1
-1
-2
0
2
-3
-3
-2
-2
3
2
0
-1
0
-2
-1
1
-2
-2
0 -1 = ADJUNTA2 A-2
-3
3
-3
2
B =
Bt
Sea A =
Cof A11 = Cof A12 =
3 1
2 2
obtener adjunta A
2
Cof A21 =
-2
-1
Cof A21 =
1
B =
0
-1
-1
1
0
-1
-1 = ADJUNTA A
1
Bt
cosΘ senΘ 0
Sea A =
-sen Θ 0
cosΘ 0 0
obtener adjunta A
1
Cof A11 =
cosΘ Cof A21 = - senΘ Cof A31 =
Cof A12 =
senΘ
Cof A13= 0
Cof A22 = cosΘ Cof A23 =
Cof A32 = 0 Cof A33 =
0 0 1
cosΘ senΘ 0
B =
-sen Θ 0
cosΘ 0 0
1
cosΘ
-sen 0 Θ senΘ cosΘ 0
Bt = ADJUNTA A 0
0
1
Sea A =
Cof A11 = Cof A12 =
0 1
1 0
obtener adjunta A
0
Cof A21 =
1
1
Cof A21 =
0
B =
0
1
1
0
0
1
1 = ADJUNTA A
0
Bt
Al calcular la adjunta de una matriz, no olvide transponer la matriz de cofactores Sea A =
=
a11 a21
a22 -a21
a12 a22
-a12 a11
Entonces adj A=
A11 A21
A12 A22
Sea A una matriz de orden n y sea B la matriz formada por los cofactores de A. Entonces la adjunta de A, que se denota adjunta A, escrito adj A, es la matriz transpuesta de la matriz B
Sea A =
2 4 3
4 6 5 6 1 -2
obtener adjunta A
1.- Primero se calculan los cofactores de cada miembro de la matriz. Cof A11 = (-1)1+15 1
6 2
= ( 5)(-2) – (1)(6) = -16
Sea A =
2 4 3
4 6 5 6 1 -2
Cof A12 = (-1)1+24 26 3
6 2
= - ( 4)(-2)- (3)(6) =
Sea A =
2 4 3
4 6 5 6 1 -2
Cof A13 = (-1)1+34 -11 3
5 1
= ( 4)(1) – (3)(5) =
Cof A21 = (-1)2+1 4 1
6 2
= -( ( 4)(-2) – (1)(6)) = 14
Cof A22 = (-1)2+2 2 3
6 2
= ( 2)(-2) – (3)(6) = -22
Cof A23 = (-1)2+3 2 3
4 1
= -( ( 2)(1) – (3)(4)) = 10
Cof A31 = (-1)3+1 4 5
6 6
= ( 4)(6) – (5)(6) = -6
Cof A32 = (-1)3+2 2 4
6 6
= -( ( 2)(6) – (4)(6)) = 12
Cof A11 = (-1)3+3 2 4
4 5
= ( 2)(5) – (4)(4) = -6
2.- Se crea la matriz B con los cofactores que se obtuvieron de la matriz A
B
=
-16
26
-11
14
-22
10
-6
12
-6
Bt
3.- Se crea la matriz B transpuesta.
=
-16
14
-6
26
-22
12
-11
-6
-6
ADJUNTA A
Sea A =
Cof A11 = =0
1
-3
0
2
3
-2
-6
-2
-1 2 10
2
5
-1
6
1
3
obtener adjunta A
-12
-2
-6
(-1)210
2
5
1
3
6
Cof A12 = -1
Cof A13 = = 0
3
-2
-6
-2
2
5
-1
1
3
(-1)
3
3
-12
-6
(-1)4-2
10
5
6
3
-1
=
Cof A14 = = 2
3
-12
-2
(-1)5-2
10
2
6
1
-1
Cof A21 = -1 Cof A22 = 1
Cof A31 =
0
Cof A41 = -2
Cof A32 =
2
Cof A42 = -2
Cof A23 = -1
Cof A33 = -3
Cof A43 =
Cof A24 =
Cof A34 = -3
Cof A44 = 2
-2
3
0
-1
0
2
-1
1
-1
-2
0
2
-3
-3
-2
-2
3
2
0
-1
0
-2
-1
1
-2
-2
0 -1 = ADJUNTA2 A-2
-3
3
-3
2
B =
Bt
Sea A =
Cof A11 = Cof A12 =
3 1
2 2
obtener adjunta A
2
Cof A21 =
-2
-1
Cof A21 =
1
B =
0
-1
-1
1
0
-1
-1 = ADJUNTA A
1
Bt
cosΘ senΘ 0
Sea A =
-sen Θ 0
cosΘ 0 0
obtener adjunta A
1
Cof A11 =
cosΘ Cof A21 = - senΘ Cof A31 =
Cof A12 =
senΘ
Cof A13= 0
Cof A22 = cosΘ Cof A23 =
Cof A32 = 0 Cof A33 =
0 0 1
cosΘ senΘ 0
B =
-sen Θ 0
cosΘ 0 0
1
cosΘ
-sen 0 Θ senΘ cosΘ 0
Bt = ADJUNTA A 0
0
1
Sea A =
Cof A11 = Cof A12 =
0 1
1 0
obtener adjunta A
0
Cof A21 =
1
1
Cof A21 =
0
B =
0
1
1
0
0
1
1 = ADJUNTA A
0
Bt
Al calcular la adjunta de una matriz, no olvide transponer la matriz de cofactores Sea A =
=
a11 a21
a22 -a21
a12 a22
-a12 a11
Entonces adj A=
A11 A21
A12 A22
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