Taller I
Páginas: 5 (1231 palabras)
Publicado: 20 de abril de 2013
Determinante de una matriz cuadrada
y Sistemas de matrices elementales
Universidad Nacional de Colombia
Facultad de Ingeniería
Programa Curricular de Ingeniería Agrícola
Juan Manuel Muñoz Penna
Código: 254112
Abril 01 de 2013
Determinantes de matrices cuadradas
Calcule el determinante
1.
1
0
2
0
1
1
1
1
≡1
3
4
0
4
0
;
0
2
-0
4
0
+3
0
2
1
1≡1(0-4)+3(0-2)≡1(-4)+3(-2)≡-4-6
|A| = -10;
−1 1 0
2 1 4 ;
1 5 6
1 4
2
≡ -1
-1
5 6
1
2.
4
6
+0
2
1
1
5
≡ -1 ( 6 - 20 ) - 1 ( 12 - 4 ) ≡ -1 ( - 14 ) -1 ( 8 ) ≡ 14 - 8
|A| = 6
3.
3
6
2
−1
3
−1
4
5
6
;
1
≡3
3
−1
5
6
+1
5
6
6
2
6
2
+4
3
−1
≡ 3 ( 18 + 5 ) + 1 ( 36 - 10 ) + 4 ( -6 - 6 ) ≡ 3( 23 ) + 1 ( - 26) + 4 ( -12 )
≡ 69 + 26 - 48
|A| = 47
−1 0 6
0 2 4
;
1 2 −3
0
2 4
-0
≡ -1
1
2 −3
4.
4
−3
+6
0
1
2
2
≡ -1 ( -6 - 8 ) + 6 ( 0 - 2 ) ≡ -1 ( -14 ) + 6 ( -2 )
≡ 14 - 12
|A| = 2
−2 3 1
4 6 5 ;
0 2 1
4
6 5
-3
≡ -2
0
2 1
5.
5
1
+1
4
0
6
2
≡ -2 ( 6 - 10 ) - 3 ( 4 - 0 ) + 1 ( 8 - 0 ) ≡ -2 ( -4 ) - 3 ( 4 ) + 1 (8)
≡ 8 - 12 + 8
|A| = 45 −2 1
6
0 3 ;
−2 1 4
6
0 3
+2
≡5
−2
1 4
6.
3
4
+1
6
−2
0
1
≡ 5 ( 0 - 3 ) + 2 ( 24 + 6 ) + 1 ( 6 - 0 ) ≡ 5 ( -3 ) + 2 ( 30 ) + 1 (6)
≡ -15 + 60 + 6
|A| = 51
2
7.
2
0
0
1
0
1
0
2
1
0
2
≡2
3
4
1
3
4
1
3
1
2
5
0
2
5
0
;
0
0
1
-0
4
1
3
2
5
0
+3
1
3
2 1
5
0
0
0
1
1
0
2
−4
2
50
0
2
+1
5
0
+2
0
0
1
1
0
2
0
2
4
1
3
1
3
+3 0
0
2
5
0
−1
0
1
5
0
+2
0
1
0
2
−1 0
0
2
1
3
−1
0
1
1
3
+4
0
1
0
2
≡ 2 [ 1 ( 0 - 15 ) - 4 ( 0 - 10 ) + 2 ( 0 - 2 ) ] + 3 [ - 1 ( 0 - 5 ) ] -1 [ - 1 ( 0 - 1 ) ]
≡ 2 [ 1 ( -15 - 4 -10 ) + 2 ( -2 ) ] + 3 [ -1 ( -5 ) ] -1 [ -1 ( -1 ) ]
≡ 2 [ 1 ( 21) ] + 3 [ 5 ] -1 [ 1 ]
≡ 2 [ 21 ] + 3 [ 5 ] -1 [ 1 ]
≡ 42 + 15 -1
|A| = 56
8.
−3
−4
5
2
≡ -3
7
8
3
0 0
7 0
8 −1
3 0
0
−1
0
0
0
0
6
0
0
6
;
-0
−4
5
2
−3 7
0
−1
0
−1
0
0
0
6
+0
0
6
−0
−4
5
2
8
3
≡ -3 [ 7 ( - 6 - 0 ) ]
≡ -3 [ 7 ( -6 ) ]
3
7 0
8 0
3 6
0
6
+0
-0
8
3
−4
5
2
−1
0
7 0
8 −1
30
≡ -3 [ -42 ]
|A| = 126
9.
−2
1
3
4
0 0
2 −1
0 −1
2 3
−1 4
−1 5
3 0
2
0
2
≡ -2
7
4
5
0
;
1
3
4
-0
−1 4
−1 5
3 0
−2 2
−1
3
−7 1
0
2
1
3
4
+0
2
0
2
4
5
0
5
0
+1
0
2
5
0
−1
3
−2
3
4
−1
3
-7
+4
1 2
3 0
4 2
−1
3
0
2
3
4
−1
−1
−1
3
0
2
≡ -2 [ 2 ( 0 - 15) + 1 ( 0 - 10 ) + 4 ( 0 + 2 ) ] - 7 [ 1 ( 0 + 2 ) - 2 ( 9 + 4 ) - 1 ( 6 - 0 ) ]
≡ -2 [ 2 ( -15 ) + 1 ( -10 ) + 4 ( 2 ) ] - 7 [ 1 ( 2 ) - 2 ( 13 ) - 1 (6) ]
≡ -2 [ -30 - 10 +8 ] - 7 [ 2 - 26 - 6 ]
≡ -2 [ -32 ] - 7 [ -30 ]
≡ 64 + 210
|A| = 274
10.
2
0
0
0
0
3
1
0
0
0
−1
7
−4
0
0
4
8
−1
−2
0
≡2
1
0
0
0
7
4
0
0
8
−1
−2
0
2
5
8
6
5
25
8
6
-3
0 7
0 4
0 0
0 0
8
−1
−2
0
2
5
8
6
1
0
0
0
-1
4
1
0
0
0
8
−1
−2
0
2
5
8
6
-4
0
0
0
0
1
0
0
0
7
4
0
0
2
5
8
6
+5
0 1 7
0 0 4
0 0 0
0 0 0
8
−1
−2
0
=2 1
−1 5
−2 8
0 6
4
0
0
−7
=2 1 4
−2
0
−1 5
−2 8
0 6
0
0
0
8
6
+8
0
0
−1
8
6
0
0
04
0
0
+5
5
8
6
0
0
+2
0
0
0
4
0
0
−1
−2
0
−2
0
≡ 2 [ 1 [ 4 ( -12 - 0 ) ] ]
≡ 2 [ 1 [ 4 ( -12 ) ] ]
≡ 8 ( -12 )
|A| = -96
11. Demuestre que si A y B son matrices diagonales de nxn, entonces |AB| = |A|.|B|
Sea A =
a11
0
0
.
.
.
0
0
a22
0
.
.
.
0
0
0
a33
.
.
.
0
...
...
...
0
0
0
....
y Sistemas de matrices elementales
Universidad Nacional de Colombia
Facultad de Ingeniería
Programa Curricular de Ingeniería Agrícola
Juan Manuel Muñoz Penna
Código: 254112
Abril 01 de 2013
Determinantes de matrices cuadradas
Calcule el determinante
1.
1
0
2
0
1
1
1
1
≡1
3
4
0
4
0
;
0
2
-0
4
0
+3
0
2
1
1≡1(0-4)+3(0-2)≡1(-4)+3(-2)≡-4-6
|A| = -10;
−1 1 0
2 1 4 ;
1 5 6
1 4
2
≡ -1
-1
5 6
1
2.
4
6
+0
2
1
1
5
≡ -1 ( 6 - 20 ) - 1 ( 12 - 4 ) ≡ -1 ( - 14 ) -1 ( 8 ) ≡ 14 - 8
|A| = 6
3.
3
6
2
−1
3
−1
4
5
6
;
1
≡3
3
−1
5
6
+1
5
6
6
2
6
2
+4
3
−1
≡ 3 ( 18 + 5 ) + 1 ( 36 - 10 ) + 4 ( -6 - 6 ) ≡ 3( 23 ) + 1 ( - 26) + 4 ( -12 )
≡ 69 + 26 - 48
|A| = 47
−1 0 6
0 2 4
;
1 2 −3
0
2 4
-0
≡ -1
1
2 −3
4.
4
−3
+6
0
1
2
2
≡ -1 ( -6 - 8 ) + 6 ( 0 - 2 ) ≡ -1 ( -14 ) + 6 ( -2 )
≡ 14 - 12
|A| = 2
−2 3 1
4 6 5 ;
0 2 1
4
6 5
-3
≡ -2
0
2 1
5.
5
1
+1
4
0
6
2
≡ -2 ( 6 - 10 ) - 3 ( 4 - 0 ) + 1 ( 8 - 0 ) ≡ -2 ( -4 ) - 3 ( 4 ) + 1 (8)
≡ 8 - 12 + 8
|A| = 45 −2 1
6
0 3 ;
−2 1 4
6
0 3
+2
≡5
−2
1 4
6.
3
4
+1
6
−2
0
1
≡ 5 ( 0 - 3 ) + 2 ( 24 + 6 ) + 1 ( 6 - 0 ) ≡ 5 ( -3 ) + 2 ( 30 ) + 1 (6)
≡ -15 + 60 + 6
|A| = 51
2
7.
2
0
0
1
0
1
0
2
1
0
2
≡2
3
4
1
3
4
1
3
1
2
5
0
2
5
0
;
0
0
1
-0
4
1
3
2
5
0
+3
1
3
2 1
5
0
0
0
1
1
0
2
−4
2
50
0
2
+1
5
0
+2
0
0
1
1
0
2
0
2
4
1
3
1
3
+3 0
0
2
5
0
−1
0
1
5
0
+2
0
1
0
2
−1 0
0
2
1
3
−1
0
1
1
3
+4
0
1
0
2
≡ 2 [ 1 ( 0 - 15 ) - 4 ( 0 - 10 ) + 2 ( 0 - 2 ) ] + 3 [ - 1 ( 0 - 5 ) ] -1 [ - 1 ( 0 - 1 ) ]
≡ 2 [ 1 ( -15 - 4 -10 ) + 2 ( -2 ) ] + 3 [ -1 ( -5 ) ] -1 [ -1 ( -1 ) ]
≡ 2 [ 1 ( 21) ] + 3 [ 5 ] -1 [ 1 ]
≡ 2 [ 21 ] + 3 [ 5 ] -1 [ 1 ]
≡ 42 + 15 -1
|A| = 56
8.
−3
−4
5
2
≡ -3
7
8
3
0 0
7 0
8 −1
3 0
0
−1
0
0
0
0
6
0
0
6
;
-0
−4
5
2
−3 7
0
−1
0
−1
0
0
0
6
+0
0
6
−0
−4
5
2
8
3
≡ -3 [ 7 ( - 6 - 0 ) ]
≡ -3 [ 7 ( -6 ) ]
3
7 0
8 0
3 6
0
6
+0
-0
8
3
−4
5
2
−1
0
7 0
8 −1
30
≡ -3 [ -42 ]
|A| = 126
9.
−2
1
3
4
0 0
2 −1
0 −1
2 3
−1 4
−1 5
3 0
2
0
2
≡ -2
7
4
5
0
;
1
3
4
-0
−1 4
−1 5
3 0
−2 2
−1
3
−7 1
0
2
1
3
4
+0
2
0
2
4
5
0
5
0
+1
0
2
5
0
−1
3
−2
3
4
−1
3
-7
+4
1 2
3 0
4 2
−1
3
0
2
3
4
−1
−1
−1
3
0
2
≡ -2 [ 2 ( 0 - 15) + 1 ( 0 - 10 ) + 4 ( 0 + 2 ) ] - 7 [ 1 ( 0 + 2 ) - 2 ( 9 + 4 ) - 1 ( 6 - 0 ) ]
≡ -2 [ 2 ( -15 ) + 1 ( -10 ) + 4 ( 2 ) ] - 7 [ 1 ( 2 ) - 2 ( 13 ) - 1 (6) ]
≡ -2 [ -30 - 10 +8 ] - 7 [ 2 - 26 - 6 ]
≡ -2 [ -32 ] - 7 [ -30 ]
≡ 64 + 210
|A| = 274
10.
2
0
0
0
0
3
1
0
0
0
−1
7
−4
0
0
4
8
−1
−2
0
≡2
1
0
0
0
7
4
0
0
8
−1
−2
0
2
5
8
6
5
25
8
6
-3
0 7
0 4
0 0
0 0
8
−1
−2
0
2
5
8
6
1
0
0
0
-1
4
1
0
0
0
8
−1
−2
0
2
5
8
6
-4
0
0
0
0
1
0
0
0
7
4
0
0
2
5
8
6
+5
0 1 7
0 0 4
0 0 0
0 0 0
8
−1
−2
0
=2 1
−1 5
−2 8
0 6
4
0
0
−7
=2 1 4
−2
0
−1 5
−2 8
0 6
0
0
0
8
6
+8
0
0
−1
8
6
0
0
04
0
0
+5
5
8
6
0
0
+2
0
0
0
4
0
0
−1
−2
0
−2
0
≡ 2 [ 1 [ 4 ( -12 - 0 ) ] ]
≡ 2 [ 1 [ 4 ( -12 ) ] ]
≡ 8 ( -12 )
|A| = -96
11. Demuestre que si A y B son matrices diagonales de nxn, entonces |AB| = |A|.|B|
Sea A =
a11
0
0
.
.
.
0
0
a22
0
.
.
.
0
0
0
a33
.
.
.
0
...
...
...
0
0
0
....
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