CAPITULO XXV
Hallar el valor de una determinante de tercer orden
Hallar el valor de una determinante de tercer orden
- Ejercicio 187
Hallar el valor de las siguientes determinantes:
- | 1 2 1 1 3 4 1 0 2 | | 1 2 1 1 3 4 1 0 2 1 2 1 1 3 4 | = [ ( 1 × 3 × 2 ) +( 1 × 0 × 1 ) +( 1 × 2 × 4 ) ] –[ ( 1 × 3 × 1 ) +( 4 × 0 × 1 ) +( 2 × 2 × 1 ) ] = [ 6+0+8 ] –[ 3+0+4 ] = 14–7 = 7
- | 1 2 – 2 1 – 3 3 – 1 4 5 | | 1 2 – 2 1 – 3 3 – 1 4 5 1 2 – 2 1 – 3 3 | = [ ( 1 × –3 × 5 ) +( 1 × 4 × –2 ) +( –1 × 2 × 3 ) ] –[ ( –2 × –3 × –1 ) +( 3 × 4 × 1 ) +( 5 × 2 × 1 ) ] = [ –15–8–6 ] –[ –6+12+10 ] = –29–16 = –45
- | – 3 4 1 2 – 3 0 1 2 7 | | – 3 4 1 2 – 3 0 1 2 7 – 3 4 1 2 – 3 0 | = [ ( –3 × –3 × 7 ) +( 2 × 2 × 1 ) +( 1 × 4 × 0 ) ] –[ ( 1 × –3 × –1 ) +( –3 × 2 × 0 ) +( 2 × 4 × 7 ) ] = [ 63+4+0 ] –[ –3+0+56 ] = 67–53 = 14
- | 2 5 – 1 3 – 4 3 6 2 4 | | 2 5 – 1 3 – 4 3 6 2 4 2 5 – 1 3 – 4 3 | = [ ( 2 × –4 × 4 ) +( 3 × 2 × –1 ) +( 6 × 5 × 3 ) ] –[ ( –1 × –4 × 6 ) +( 2 × 2 × 3 ) +( 3 × 5 × 4 ) ] = [ –32–6+90 ] –[ 24+12+60 ] = 52–96 = –44
- | 5 – 1 – 6 – 2 5 3 3 4 2 | | 5 – 1 – 6 – 2 5 3 3 4 2 5 – 1 – 6 – 2 5 3 | = [ ( 5 × 5 × 2 ) +( –2 × 4 × –6 ) +( 3 × –1 × 3 ) ] –[ ( 3 × 5 × –6 ) +( 5 × 4 × 3 ) +( –2 × –1 × 2 ) ] = [ 50+48–9 ] –[ –90+60+4 ] = 89+26 = 115
- | 4 1 5 3 2 – 6 12 3 2 | | 4 1 5 3 2 – 6 12 3 2 4 1 5 3 2 – 6 | = [ ( 4 × 2 × 2 ) +( 3 × 3 × 5 ) +( 12 × 1 × –6 ) ] –[ ( 12 × 2 × 5 ) +( 4 × 3 × –6 ) +( 3 × 1 × 2 ) ] = [ 16+45–72 ] –[ 120–72+6 ] = –11–54 = –65
- | 5 2 – 8 – 3 – 7 3 4 0 – 1 | | 5 2 – 8 – 3 – 7 3 4 0 – 1 5 2 – 8 – 3 – 7 3 | = [ ( 5 × –7 × –1 ) +( –3 × 0 × –8 ) +( 4 × 2 × 3 ) ] –[ ( 4 × –7 × –8 ) +( 5 × 0 × 3 ) +( –3 × 2 × –1 ) ] = [ 35+0+24 ] –[ 224+0+6 ] = 59–230 = –171
- | 3 2 5 – 1 – 3 4 3 2 5 | | 3 2 5 – 1 – 3 4 3 2 5 3 2 5 – 1 – 3 4 | = [ ( 3 × –3 × 5 ) +( –1 × 2 × 5 ) +( 3 × 2 × 4 ) ] –[ ( 3 × –3 × 5 ) +( 3 × 2 × 4 ) +( –1 × 2 × 5 ) ] = [ –45–10+24 ] –[ –45+24–10 ] = –31+31 = 0
- | 5 2 3 6 1 2 3 4 5 | | 5 2 3 6 1 2 3 4 5 5 2 3 6 1 2 | = [ ( 5 × 1 × 5 ) +( 6 × 4 × 3 ) +( 3 × 2 × 2 ) ] –[ ( 3 × 1 × 3 ) +( 5 × 4 × 2 ) +( 6 × 2 × 5 ) ] = [ 25+72+12 ] –[ 9+40+60 ] = 109–109 = 0
- | 12 5 10 8 – 6 9 7 4 – 2 | | 12 5 10 8 – 6 9 7 4 – 2 12 5 10 8 – 6 9 | = [ ( 12 × –6 × –2 ) +( 8 × 4 × 10 ) +( 7 × 5 × 9 ) ] –[ ( 7 × –6 × 10 ) +( 12 × 4 × 9 ) +( 8 × 5 × –2 ) ] = [ 144+320+315 ] –[ –420+432–80 ] = 779+68 = 847
- | – 9 3 – 4 7 – 5 – 3 4 6 1 | | – 9 3 – 4 7 – 5 – 3 4 6 1 – 9 3 – 4 7 – 5 – 3 | = [ ( –9 × –5 × 1 ) +( 7 × 6 × –4 ) +( 4 × 3 × –3 ) ] –[ ( 4 × –5 × –4 ) +( –9 × 6 × –3 ) +( 7 × 3 × 1 ) ] = [ 45–168–36 ] –[ 80+162+21 ] = –159–263 = –422
- | 11 – 5 7 – 12 3 8 – 13 1 9 | | 11 – 5 7 – 12 3 8 – 13 1 9 11 – 5 7 – 12 3 8 | = [ ( 11 × 3 × 9 ) +( –12 × 1 × 7 ) +( –13 × –5 × 8 ) ] –[ ( –13 × 3 × 7 ) +( 11 × 1 × 8 ) +( –12 × –5 × 9 ) ] = [ 297–84+520 ] –[ –273+88+540 ] = 733–355 = 378