Icono del sitio Solucionario Baldor

Ejercicio 188

Gestor Baldor
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DESCARGA
CAPITULO XXV

Ecuaciones simultáneas con tres incógnitas
Resolución por determinantes
Ejercicio 188
Resolver por determinantes:
  1. { x+y+z=11 x–y+3z=13 2x+2y–z=7 x = | 11 1 1 13 – 1 3 7 2 – 1 | | 1 1 1 1 – 1 3 2 2 – 1 | = | 11 1 1 13 – 1 3 7 2 – 1 11 1 1 13 – 1 3 | | 1 1 1 1 – 1 3 2 2 – 1 1 1 1 1 – 1 3 | = [ ( 11 × –1 × –1 ) +( 13 × 2 × 1 ) +( 7 × 1 × 3 ) ] –[ ( 7 × –1 × 1 ) +( 11 × 2 × 3 ) +( 1 × 1 × –13 ) ] [ ( 1 × –1 × –1 ) +( 1 × 2 × 1 ) +( 2 × 1 × 3 ) ] –[ ( 2 × –1 × 1 ) +( 1 × 2 × 3 ) +( 1 × 1 × –1 ) ] = [ 11+26+21 ] –[ –7+66–13 ] [ 1+2+6 ] –[ –2+6–1 ] = 58–46 9–3 = 6 = 2 y = | 1 11 1 1 13 3 2 7 – 1 | | 1 1 1 1 – 1 3 2 2 – 1 | = | 1 11 1 1 13 3 2 7 – 1 1 11 1 1 13 3 | | 1 1 1 1 – 1 3 2 2 – 1 1 1 1 1 – 1 3 | = [ ( 1 × 13 × –1 ) +( 1 × 7 × 1 ) +( 2 × 11 × 3 ) ] –[ ( 2 × 13 × 1 ) +( 1 × 7 × 3 ) +( 1 × 11 × –1 ) ] [ ( 1 × –1 × –1 ) +( 1 × 2 × 1 ) +( 2 × 1 × 3 ) ] –[ ( 2 × –1 × 1 ) +( 1 × 2 × 3 ) +( 1 × 1 × –1 ) ] = [ –13+7+66 ] –[ 26+21–11 ] [ 1+2+6 ] –[ –2+6–1 ] = 60–36 9–3 = 6 = 4 z = | 1 1 11 1 – 1 13 2 2 7 | | 1 1 1 1 – 1 3 2 2 – 1 | = | 1 1 11 1 – 1 13 2 2 7 1 1 11 1 – 1 13 | | 1 1 1 1 – 1 3 2 2 – 1 1 1 1 1 – 1 3 | = [ ( 1 × –1 × 7 ) +( 1 × 2 × 11 ) +( 2 × 1 × 13 ) ] –[ ( 2 × –1 × 11 ) +( 1 × 2 × 13 ) +( 1 × 1 × 7 ) ] [ ( 1 × –1 × –1 ) +( 1 × 2 × 1 ) +( 2 × 1 × 3 ) ] –[ ( 2 × –1 × 1 ) +( 1 × 2 × 3 ) +( 1 × 1 × –1 ) ] = [ –7+22+26 ] –[ –22+26+7 ] [ 1+2+6 ] –[ –2+6–1 ] = 41–11 9–3 = 6 = 5 Sol.{ x=2 y=4 z=5
  2. { x+y+z=–6 2x+y–z=–1 x–2y+3z=–6 x = | –6 1 1 –1 1 – 1 –6 – 2 3 | | 1 1 1 2 1 – 1 1 – 2 3 | = | –6 1 1 –1 1 – 1 –6 – 2 3 –6 1 1 –1 1 – 1 | | 1 1 1 2 1 – 1 1 – 2 3 1 1 1 2 1 – 1 | = [ ( –6 × 1 × 3 ) +( –1 × –2 × 1 ) +( –6 × 1 × –1 ) ] –[ ( –6 × 1 × 1 ) +( –6 × –2 × –1 ) +( –1 × 1 × 3 ) ] [ ( 1 × 1 × 3 ) +( 2 × –2 × 1 ) +( 1 × 1 × –1 ) ] –[ ( 1 × 1 × 1 ) +( 1 × –2 × –1 ) +( 2 × 1 × 3 ) ] = [ –18+2+6 ] –[ –6–12–3 ] [ 3–4–1 ] –[ 1+2+6 ] = –10+21 –2–9 = – 11 11 = –1 y = | 1 – 6 1 2 – 1 – 1 1 – 6 3 | | 1 1 1 2 1 – 1 1 – 2 3 | = | 1 – 6 1 2 – 1 – 1 1 – 6 3 1 – 6 1 2 – 1 – 1 | | 1 1 1 2 1 – 1 1 – 2 3 1 1 1 2 1 – 1 | = [ ( 1 × –1 × 3 ) +( 2 × –6 × 1 ) +( 1 × –6 × –1 ) ] –[ ( 1 × –1 × 1 ) +( 1 × –6 × –1 ) +( 2 × –6 × 3 ) ] [ ( 1 × 1 × 3 ) +( 2 × –2 × 1 ) +( 1 × 1 × –1 ) ] –[ ( 1 × 1 × 1 ) +( 1 × –2 × –1 ) +( 2 × 1 × 3 ) ] = [ –3–12+6 ] –[ –1+6–36 ] [ 3–4–1 ] –[ 1+2+6 ] = –9+31 –2–9 = – 11 = –2 z = | 1 1 – 6 2 1 – 1 1 – 2 – 6 | | 1 1 1 2 1 – 1 1 – 2 3 | = | 1 1 – 6 2 1 – 1 1 – 2 – 6 1 1 – 6 2 1 – 1 | | 1 1 1 2 1 – 1 1 – 2 3 1 1 1 2 1 – 1 | = [ ( 1 × 1 × –6 ) +( 2 × –2 × –6 ) +( 1 × 1 × –1 ) ] –[ ( 1 × 1 × –6 ) +( 1 × –2 × –1 ) +( 2 × 1 × –6 ) ] [ ( 1 × 1 × 3 ) +( 2 × –2 × 1 ) +( 1 × 1 × –1 ) ] –[ ( 1 × 1 × 1 ) +( 1 × –2 × –1 ) +( 2 × 1 × 3 ) ] = [ –6+24–1 ] –[ –6+2–12 ] [ 3–4–1 ] –[ 1+2+6 ] = 17+16 –2–9 = – 11 = –3 Sol.{ x=–1 y=–2 z=–3
  3. { 2x+3y+4z=3 2x+6y+8z=5 4x+9y–4z=4 x = | 3 3 4 5 6 8 4 9 – 4 | | 2 3 4 2 6 8 4 9 – 4 | = | 3 3 4 5 6 8 4 9 – 4 3 3 4 5 6 8 | | 2 3 4 2 6 8 4 9 – 4 2 3 4 2 6 8 | = [ ( 3 × 6 × –4 ) +( 5 × 9 × 4 ) +( 4 × 3 × 8 ) ] –[ ( 4 × 6 × 4 ) +( 3 × 9 × 8 ) +( 5 × 3 × –4 ) ] [ ( 2 × 6 × –4 ) +( 2 × 9 × 4 ) +( 4 × 3 × 8 ) ] –[ ( 4 × 6 × 4 ) +( 2 × 9 × 8 ) +( 2 × 3 × –4 ) ] = [ –72+180+96 ] –[ 96+216–60 ] [ –48+72+96 ] –[ 96+144–24 ] = 204–252 120–216 = – 48 – = 1 2 y = | 2 3 4 2 5 8 4 4 – 4 | | 2 3 4 2 6 8 4 9 – 4 | = | 2 3 4 2 5 8 4 4 – 4 2 3 4 2 5 8 | | 2 3 4 2 6 8 4 9 – 4 2 3 4 2 6 8 | = [ ( 2 × 5 × –4 ) +( 2 × 4 × 4 ) +( 4 × 3 × 8 ) ] –[ ( 4 × 5 × 4 ) +( 2 × 4 × 8 ) +( 2 × 3 × –4 ) ] [ ( 2 × 6 × –4 ) +( 2 × 9 × 4 ) +( 4 × 3 × 8 ) ] –[ ( 4 × 6 × 4 ) +( 2 × 9 × 8 ) +( 2 × 3 × –4 ) ] = [ –40+32+96 ] –[ 80+64–24 ] [ –48+72+96 ] –[ 96+144–24 ] = 88–120 120–216 = – 32 – = 1 3 z = | 2 3 3 2 6 5 4 9 4 | | 2 3 4 2 6 8 4 9 – 4 | = | 2 3 3 2 6 5 4 9 4 2 3 3 2 6 5 | | 2 3 4 2 6 8 4 9 – 4 2 3 4 2 6 8 | = [ ( 2 × 6 × 4 ) +( 2 × 9 × 3 ) +( 4 × 3 × 5 ) ] –[ ( 4 × 6 × 3 ) +( 2 × 9 × 5 ) +( 2 × 3 × 4 ) ] [ ( 2 × 6 × –4 ) +( 2 × 9 × 4 ) +( 4 × 3 × 8 ) ] –[ ( 4 × 6 × 4 ) +( 2 × 9 × 8 ) +( 2 × 3 × –4 ) ] = [ 48+54+60 ] –[ 72+90+24 ] [ –48+72+96 ] –[ 96+144–24 ] = 162–186 120–216 = – 24 – = 1 4 Sol.{ x= 1 2 y= 1 3 z= 1 4
  4. { 4x–y+z=4 2y–z+2x=2 ↔ 6x+3z–2y=12 { 4x–y+z=4 2x+2y–z=2 6x–2y+3z=12 x = | 4 – 1 1 2 2 – 1 12 – 2 3 | | 4 – 1 1 2 2 – 1 6 – 2 3 | = | 4 – 1 1 2 2 – 1 12 – 2 3 4 – 1 1 2 2 – 1 | | 4 – 1 1 2 2 – 1 6 – 2 3 4 – 1 1 2 2 – 1 | = [ ( 4 × 2 × 3 ) +( 2 × –2 × 1 ) +( 12 × –1 × –1 ) ] –[ ( 12 × 2 × 1 ) +( 4 × –2 × –1 ) +( 2 × –1 × 3 ) ] [ ( 4 × 2 × 3 ) +( 2 × –2 × 1 ) +( 6 × –1 × –1 ) ] –[ ( 6 × 2 × 1 ) +( 4 × –2 × –1 ) +( 2 × –1 × 3 ) ] = [ 24–4+12 ] –[ 24+8–6 ] [ 24–4+6 ] –[ 12+8–6 ] = 32–26 26–14 = 6 = 1 2 y = | 4 4 1 2 2 – 1 6 12 3 | | 4 – 1 1 2 2 – 1 6 – 2 3 | = | 4 4 1 2 2 – 1 6 12 3 4 4 1 2 2 – 1 | | 4 – 1 1 2 2 – 1 6 – 2 3 4 – 1 1 2 2 – 1 | = [ ( 4 × 2 × 3 ) +( 2 × 12 × 1 ) +( 6 × 4 × –1 ) ] –[ ( 6 × 2 × 1 ) +( 4 × 12 × –1 ) +( 2 × 4 × 3 ) ] [ ( 4 × 2 × 3 ) +( 2 × –2 × 1 ) +( 6 × –1 × –1 ) ] –[ ( 6 × 2 × 1 ) +( 4 × –2 × –1 ) +( 2 × –1 × 3 ) ] = [ 24+24–24 ] –[ 12–48+24 ] [ 24–4+6 ] –[ 12+8–6 ] = 24+12 26–14 = 12 = 3 z = | 4 – 1 4 2 2 2 6 – 2 12 | | 4 – 1 1 2 2 – 1 6 – 2 3 | = | 4 – 1 4 2 2 2 6 – 2 12 4 – 1 4 2 2 2 | | 4 – 1 1 2 2 – 1 6 – 2 3 4 – 1 1 2 2 – 1 | = [ ( 4 × 2 × 12 ) +( 2 × –2 × 4 ) +( 6 × –1 × 2 ) ] –[ ( 6 × 2 × 4 ) +( 4 × –2 × 2 ) +( 2 × –1 × 12 ) ] [ ( 4 × 2 × 3 ) +( 2 × –2 × 1 ) +( 6 × –1 × –1 ) ] –[ ( 6 × 2 × 1 ) +( 4 × –2 × –1 ) +( 2 × –1 × 3 ) ] = [ 96–16–12 ] –[ 48–16–24 ] [ 24–4+6 ] –[ 12+8–6 ] = 68–8 26–14 = 12 = 5 Sol.{ x= 1 2 y=3 z=5
  5. { x+4y+5z=11 3x–2y+z=5 4x+y–3z=–26 x = | 11 4 5 5 – 2 1 – 26 1 – 3 | | 1 4 5 3 – 2 1 4 1 – 3 | = | 11 4 5 5 – 2 1 – 26 1 – 3 11 4 5 5 – 2 1 | | 1 4 5 3 – 2 1 4 1 – 3 1 4 5 3 – 2 1 | = [ ( 11 × –2 × –3 ) +( 5 × 1 × 5 ) +( –26 × 4 × 1 ) ] –[ ( –26 × –2 × 5 ) +( 11 × 1 × 1 ) +( 5 × 4 × –3 ) ] [ ( 1 × –2 × –3 ) +( 3 × 1 × 5 ) +( 4 × 4 × 1 ) ] –[ ( 4 × –2 × 5 ) +( 1 × 1 × 1 ) +( 3 × 4 × –3 ) ] = [ 66+25–104 ] –[ 260+11–60 ] [ 6+15+16 ] –[ –40+1–36 ] = –13–211 37+75 = – 112 = –2 y = | 1 11 5 3 5 1 4 – 26 – 3 | | 1 4 5 3 – 2 1 4 1 – 3 | = | 1 11 5 3 5 1 4 – 26 – 3 1 11 5 3 5 1 | | 1 4 5 3 – 2 1 4 1 – 3 1 4 5 3 – 2 1 | = [ ( 1 × 5 × –3 ) +( 3 × –26 × 5 ) +( 4 × 11 × 1 ) ] –[ ( 4 × 5 × 5 ) +( 1 × –26 × 1 ) +( 3 × 11 × –3 ) ] [ ( 1 × –2 × –3 ) +( 3 × 1 × 5 ) +( 4 × 4 × 1 ) ] –[ ( 4 × –2 × 5 ) +( 1 × 1 × 1 ) +( 3 × 4 × –3 ) ] = [ –15–390+44 ] –[ 100–26–99 ] [ 6+15+16 ] –[ –40+1–36 ] = –361+25 37+75 = – 112 = –3 z = | 1 4 11 3 – 2 5 4 1 – 26 | | 1 4 5 3 – 2 1 4 1 – 3 | = | 1 4 11 3 – 2 5 4 1 – 26 1 4 11 3 – 2 5 | | 1 4 5 3 – 2 1 4 1 – 3 1 4 5 3 – 2 1 | = [ ( 1 × –2 × –26 ) +( 3 × 1 × 11 ) +( 4 × 4 × 5 ) ] –[ ( 4 × –2 × 11 ) +( 1 × 1 × 5 ) +( 3 × 4 × –26 ) ] [ ( 1 × –2 × –3 ) +( 3 × 1 × 5 ) +( 4 × 4 × 1 ) ] –[ ( 4 × –2 × 5 ) +( 1 × 1 × 1 ) +( 3 × 4 × –3 ) ] = [ 52+33+80 ] –[ –88+5–312 ] [ 6+15+16 ] –[ –40+1–36 ] = 165+395 37+75 = 112 = 5 Sol.{ x=–2 y=–3 z=5
  6. { 7x+10y+4z=–2 5x–2y+6z=38 3x+y–z=21 x = | – 2 10 4 38 – 2 6 21 1 – 1 | | 7 10 4 5 – 2 6 3 1 – 1 | = | – 2 10 4 38 – 2 6 21 1 – 1 – 2 10 4 38 – 2 6 | | 7 10 4 5 – 2 6 3 1 – 1 7 10 4 5 – 2 6 | = [ ( –2 × –2 × –1 ) +( 38 × 1 × 4 ) +( 21 × 10 × 6 ) ] –[ ( 21 × –2 × 4 ) +( –2 × 1 × 6 ) +( 38 × 10 × –1 ) ] [ ( 7 × –2 × –1 ) +( 5 × 1 × 4 ) +( 3 × 10 × 6 ) ] –[ ( 3 × –2 × 4 ) +( 7 × 1 × 6 ) +( 5 × 10 × –1 ) ] = [ –4+152+1260 ] –[ –168–12–380 ] [ 14+20+180 ] –[ –24+42–50 ] = 1408+560 214+32 = 246 = 8 y = | 7 – 2 4 5 38 6 3 21 – 1 | | 7 10 4 5 – 2 6 3 1 – 1 | = | 7 – 2 4 5 38 6 3 21 – 1 7 – 2 4 5 38 6 | | 7 10 4 5 – 2 6 3 1 – 1 7 10 4 5 – 2 6 | = [ ( 7 × 38 × –1 ) +( 5 × 21 × 4 ) +( 3 × –2 × 6 ) ] –[ ( 3 × 38 × 4 ) +( 7 × 21 × 6 ) +( 5 × –2 × –1 ) ] [ ( 7 × –2 × –1 ) +( 5 × 1 × 4 ) +( 3 × 10 × 6 ) ] –[ ( 3 × –2 × 4 ) +( 7 × 1 × 6 ) +( 5 × 10 × –1 ) ] = [ –266+420–36 ] –[ 456+882+10 ] [ 14+20+180 ] –[ –24+42–50 ] = 118–1348 214+32 = – 246 = –5 z = | 7 10 – 2 5 – 2 38 3 1 21 | | 7 10 4 5 – 2 6 3 1 – 1 | = | 7 10 – 2 5 – 2 38 3 1 21 7 10 – 2 5 – 2 38 | | 7 10 4 5 – 2 6 3 1 – 1 7 10 4 5 – 2 6 | = [ ( 7 × –2 × 21 ) +( 5 × 1 × –2 ) +( 3 × 10 × 38 ) ] –[ ( 3 × –2 × –2 ) +( 7 × 1 × 38 ) +( 5 × 10 × 21 ) ] [ ( 7 × –2 × –1 ) +( 5 × 1 × 4 ) +( 3 × 10 × 6 ) ] –[ ( 3 × –2 × 4 ) +( 7 × 1 × 6 ) +( 5 × 10 × –1 ) ] = [ –294–10+1140 ] –[ 12+266+1050 ] [ 14+20+180 ] –[ –24+42–50 ] = 836–1328 214+32 = – 246 = –2 Sol.{ x=8 y=–5 z=–2
  7. { 4x+7y+5z=–2 6x+3y+7z=6 x–y+9z=–21 x = | – 2 7 5 6 3 7 – 21 – 1 9 | | 4 7 5 6 3 7 1 – 1 9 | = | – 2 7 5 6 3 7 – 21 – 1 9 – 2 7 5 6 3 7 | | 4 7 5 6 3 7 1 – 1 9 4 7 5 6 3 7 | = [ ( –2 × 3 × 9 ) +( 6 × –1 × 5 ) +( –21 × 7 × 7 ) ] –[ ( –21 × 3 × 5 ) +( –2 × –1 × 7 ) +( 6 × 7 × 9 ) ] [ ( 4 × 3 × 9 ) +( 6 × –1 × 5 ) +( 1 × 7 × 7 ) ] –[ ( 1 × 3 × 5 ) +( 4 × –1 × 7 ) +( 6 × 7 × 9 ) ] = [ –54–30–1029 ] –[ –315+14+378 ] [ 108–30+49 ] –[ 15–28+378 ] = –1113–77 127–365 = – – 238 = 5 y = | 4 – 2 5 6 6 7 1 – 21 9 | | 4 7 5 6 3 7 1 – 1 9 | = | 4 – 2 5 6 6 7 1 – 21 9 4 – 2 5 6 6 7 | | 4 7 5 6 3 7 1 – 1 9 4 7 5 6 3 7 | = [ ( 4 × 6 × 9 ) +( 6 × –21 × 5 ) +( 1 × –2 × 7 ) ] –[ ( 1 × 6 × 5 ) +( 4 × –21 × 7 ) +( 6 × –2 × 9 ) ] [ ( 4 × 3 × 9 ) +( 6 × –1 × 5 ) +( 1 × 7 × 7 ) ] –[ ( 1 × 3 × 5 ) +( 4 × –1 × 7 ) +( 6 × 7 × 9 ) ] = [ 216–630–14 ] –[ 30–588–108 ] [ 108–30+49 ] –[ 15–28+378 ] = –452+666 127–365 = 238 – 238 = –1 z = | 4 7 – 2 6 3 6 1 – 1 – 21 | | 4 7 5 6 3 7 1 – 1 9 | = | 4 7 – 2 6 3 6 1 – 1 – 21 4 7 – 2 6 3 6 | | 4 7 5 6 3 7 1 – 1 9 4 7 5 6 3 7 | = [ ( 4 × 3 × –21 ) +( 6 × –1 × –2 ) +( 1 × 7 × 6 ) ] –[ ( 1 × 3 × –2 ) +( 4 × –1 × 6 ) +( 6 × 7 × –21 ) ] [ ( 4 × 3 × 9 ) +( 6 × –1 × 5 ) +( 1 × 7 × 7 ) ] –[ ( 1 × 3 × 5 ) +( 4 × –1 × 7 ) +( 6 × 7 × 9 ) ] = [ –252+12+42 ] –[ –6–24–882 ] [ 108–30+49 ] –[ 15–28+378 ] = –198+912 127–365 = – 238 = –3 Sol.{ x=5 y=–1 z=–3
  8. { 3x–5y+2z=–22 2x–y+6z=32 8x+3y–5z=–33 x = | – 22 – 5 2 32 – 1 6 – 33 3 – 5 | | 3 – 5 2 2 – 1 6 8 3 – 5 | = | – 22 – 5 2 32 – 1 6 – 33 3 – 5 – 22 – 5 2 32 – 1 6 | | 3 – 5 2 2 – 1 6 8 3 – 5 3 – 5 2 2 – 1 6 | = [ ( –22 × –1 × –5 ) +( 33 × 3 × 2 ) +( –33 × –5 × 6 ) ] –[ ( –33 × –1 × 2 ) +( –22 × 3 × 6 ) +( 32 × –5 × –5 ) ] [ ( 3 × –1 × –5 ) +( 2 × 3 × 2 ) +( 8 × –5 × 6 ) ] –[ ( 8 × –1 × 2 ) +( 3 × 3 × 6 ) +( 2 × –5 × –5 ) ] = [ –110+192+990 ] –[ 66–396+800 ] [ 15+12–240 ] –[ –16+54+50 ] = 1072–470 –213–88 = – 301 = –2 y = | 3 – 22 2 2 32 6 8 – 33 – 5 | | 3 – 5 2 2 – 1 6 8 3 – 5 | = | 3 – 22 2 2 32 6 8 – 33 – 5 3 – 22 2 2 32 6 | | 3 – 5 2 2 – 1 6 8 3 – 5 3 – 5 2 2 – 1 6 | = [ ( 3 × 32 × –5 ) +( 2 × –33 × 2 ) +( 8 × –22 × 6 ) ] –[ ( 8 × 32 × 2 ) +( 3 × –33 × 6 ) +( 2 × –22 × –5 ) ] [ ( 3 × –1 × –5 ) +( 2 × 3 × 2 ) +( 8 × –5 × 6 ) ] –[ ( 8 × –1 × 2 ) +( 3 × 3 × 6 ) +( 2 × –5 × –5 ) ] = [ –480–132–1056 ] –[ 512–594+220 ] [ 15+12–240 ] –[ –16+54+50 ] = –1668–138 –213–88 = – – 301 = 6 z = | 3 – 5 – 22 2 – 1 32 8 3 – 33 | | 3 – 5 2 2 – 1 6 8 3 – 5 | = | 3 – 5 – 22 2 – 1 32 8 3 – 33 3 – 5 – 22 2 – 1 32 | | 3 – 5 2 2 – 1 6 8 3 – 5 3 – 5 2 2 – 1 6 | = [ ( 3 × –1 × –33 ) +( 2 × 3 × –22 ) +( 8 × –5 × 32 ) ] –[ ( 8 × –1 × –22 ) +( 3 × 3 × 32 ) +( 2 × –5 × –33 ) ] [ ( 3 × –1 × –5 ) +( 2 × 3 × 2 ) +( 8 × –5 × 6 ) ] –[ ( 8 × –1 × 2 ) +( 3 × 3 × 6 ) +( 2 × –5 × –5 ) ] = [ 99–132–1280 ] –[ 176+288+330 ] [ 15+12–240 ] –[ –16+54+50 ] = –1313–795 –213–88 = – – 301 = 7 Sol.{ x=–2 y=6 z=7
  9. { x+y+z=3 x+2y=6 2x+3y=6 x = | 3 1 1 6 2 0 6 3 0 | | 1 1 1 1 2 0 2 3 0 | = | 3 1 1 6 2 0 6 3 0 3 1 1 6 2 0 | | 1 1 1 1 2 0 2 3 0 1 1 1 1 2 0 | = [ ( 3 × 2 × 0 ) +( 6 × 3 × 1 ) +( 6 × 1 × 0 ) ] –[ ( 6 × 2 × 1 ) +( 3 × 3 × 0 ) +( 6 × 1 × 0 ) ] [ ( 1 × 2 × 0 ) +( 1 × 3 × 1 ) +( 2 × 1 × 0 ) ] –[ ( 2 × 2 × 1 ) +( 1 × 3 × 0 ) +( 1 × 1 × 0 ) ] = [ 0+18+0 ] –[ 12–0+0 ] [ 0+3–0 ] –[ 4+0+0 ] = 18–12 3–4 = 6 –1 = –6 y = | 1 3 1 1 6 0 2 6 0 | | 1 1 1 1 2 0 2 3 0 | = | 1 3 1 1 6 0 2 6 0 1 3 1 1 6 0 | | 1 1 1 1 2 0 2 3 0 1 1 1 1 2 0 | = [ ( 1 × 6 × 0 ) +( 1 × 6 × 1 ) +( 2 × 3 × 0 ) ] –[ ( 2 × 6 × 1 ) +( 1 × 6 × 0 ) +( 1 × 3 × 0 ) ] [ ( 1 × 2 × 0 ) +( 1 × 3 × 1 ) +( 2 × 1 × 0 ) ] –[ ( 2 × 2 × 1 ) +( 1 × 3 × 0 ) +( 1 × 1 × 0 ) ] = [ 0+6+0 ] –[ 12–0+0 ] [ 0+3–0 ] –[ 4+0+0 ] = 6–12 3–4 = –6 –1 = 6 z = | 1 1 3 1 2 6 2 3 6 | | 1 1 1 1 2 0 2 3 0 | = | 1 1 3 1 2 6 2 3 6 1 1 3 1 2 6 | | 1 1 1 1 2 0 2 3 0 1 1 1 1 2 0 | = [ ( 1 × 2 × 6 ) +( 1 × 3 × 3 ) +( 2 × 1 × 6 ) ] –[ ( 2 × 2 × 3 ) +( 1 × 3 × 6 ) +( 1 × 1 × 6 ) ] [ ( 1 × 2 × 0 ) +( 1 × 3 × 1 ) +( 2 × 1 × 0 ) ] –[ ( 2 × 2 × 1 ) +( 1 × 3 × 0 ) +( 1 × 1 × 0 ) ] = [ 12+9+12 ] –[ 12+18+6 ] [ 0+3–0 ] –[ 4+0+0 ] = 33–36 3–4 = –3 –1 = 3 Sol.{ x=–6 y=6 z=3
  10. { 3x–2y=–1 4x+z=–28 x+2y+3z=–43 x = | – 1 – 2 0 – 28 0 1 – 43 2 3 | | 3 – 2 0 4 0 1 1 2 3 | = | – 1 – 2 0 – 28 0 1 – 43 2 3 – 1 – 2 0 – 28 0 1 | | 3 – 2 0 4 0 1 1 2 3 3 – 2 0 4 0 1 | = [ ( –1 × 0 × 3 ) +( –28 × 2 × 0 ) +( –43 × –2 × 1 ) ] –[ ( –43 × 0 × 0 ) +( –1 × 2 × 1 ) +( –28 × –2 × 3 ) ] [ ( 3 × 0 × 3 ) +( 4 × 2 × 0 ) +( 1 × –2 × 1 ) ] –[ ( 1 × 0 × 0 ) +( 3 × 2 × 1 ) +( 4 × –2 × 3 ) ] = [ 0+0+86 ] –[ 0–2+168 ] [ 0+0–2 ] –[ 0+6–24 ] = 86–166 –2+18 = – 16 = –5 y = | 3 – 1 0 4 – 28 1 1 – 43 3 | | 3 – 2 0 4 0 1 1 2 3 | = | 3 – 1 0 4 – 28 1 1 – 43 3 3 – 1 0 4 – 28 1 | | 3 – 2 0 4 0 1 1 2 3 3 – 2 0 4 0 1 | = [ ( 3 × –28 × 3 ) +( 4 × –43 × 0 ) +( 1 × –1 × 1 ) ] –[ ( 1 × –28 × 0 ) +( 3 × –43 × 1 ) +( 4 × –1 × 3 ) ] [ ( 3 × 0 × 3 ) +( 4 × 2 × 0 ) +( 1 × –2 × 1 ) ] –[ ( 1 × 0 × 0 ) +( 3 × 2 × 1 ) +( 4 × –2 × 3 ) ] = [ –252+0–1 ] –[ 0–129–12 ] [ 0+0–2 ] –[ 0+6–24 ] = –253+141 –2+18 = – 16 = –7 z = | 3 – 2 – 1 4 0 – 28 1 2 – 43 | | 3 – 2 0 4 0 1 1 2 3 | = | 3 – 2 – 1 4 0 – 28 1 2 – 43 3 – 2 – 1 4 0 – 28 | | 3 – 2 0 4 0 1 1 2 3 3 – 2 0 4 0 1 | = [ ( 3 × 0 × –43 ) +( 4 × 2 × –1 ) +( 1 × –2 × –28 ) ] –[ ( 1 × 0 × –1 ) +( 3 × 2 × –28 ) +( 4 × –2 × –43 ) ] [ ( 3 × 0 × 3 ) +( 4 × 2 × 0 ) +( 1 × –2 × 1 ) ] –[ ( 1 × 0 × 0 ) +( 3 × 2 × 1 ) +( 4 × –2 × 3 ) ] = [ 0–8+56 ] –[ 0–168+344 ] [ 0+0–2 ] –[ 0+6–24 ] = 48–176 –2+18 = – 16 = –8 Sol.{ x=–5 y=–7 z=–8
  11. { x 3 – y 4 + z 4 =1 x 6 + y 2 –z=1 x 2 – y 8 – z 2 =0 x = | 1 – 1 4 1 4 1 1 2 – 1 0 – 1 8 – 1 2 | | 1 3 – 1 4 1 4 1 6 1 2 – 1 1 2 – 1 8 – 1 2 | = | 1 – 1 4 1 4 1 1 2 – 1 0 – 1 8 – 1 2 1 – 1 4 1 4 1 1 2 – 1 | | 1 3 – 1 4 1 4 1 6 1 2 – 1 1 2 – 1 8 – 1 2 1 3 – 1 4 1 4 1 6 1 2 – 1 | = [ ( 1 × 1 2 × – 1 2 ) +( 1 × – 1 8 × 1 4 ) +( 0 × – 1 4 × –1 ) ] –[ ( 0 × 1 2 × 1 4 ) +( 1 × – 1 8 × –1 ) +( 1 × – 1 4 × – 1 2 ) ] [ ( 1 3 × 1 2 × – 1 2 ) +( 1 6 × – 1 8 × 1 4 ) +( 1 2 × – 1 4 × –1 ) ] –[ ( 1 2 × 1 2 × 1 4 ) +( 1 3 × – 1 8 × –1 ) +( 1 6 × – 1 4 × – 1 2 ) ] = [ – 1 4 – 1 32 +0 ] –[ 0+ 1 8 + 1 8 ] [ – 1 12 – 1 192 + 1 8 ] –[ 1 16 + 1 24 + 1 48 ] = – 9 32 – 1 4 7 192 – 1 8 = – 17 32 – 17 = 6 y = | 1 3 1 1 4 1 6 1 – 1 1 2 0 – 1 2 | | 1 3 – 1 4 1 4 1 6 1 2 – 1 1 2 – 1 8 – 1 2 | = | 1 3 1 1 4 1 6 1 – 1 1 2 0 – 1 2 1 3 1 1 4 1 6 1 – 1 | | 1 3 – 1 4 1 4 1 6 1 2 – 1 1 2 – 1 8 – 1 2 1 3 – 1 4 1 4 1 6 1 2 – 1 | = [ ( 1 3 × 1 × – 1 2 ) +( 1 6 × 0 × 1 4 ) +( 1 2 × 1 × –1 ) ] –[ ( 1 2 × 1 × 1 4 ) +( 1 3 × 0 × –1 ) +( 1 6 × 1 × – 1 2 ) ] [ ( 1 3 × 1 2 × – 1 2 ) +( 1 6 × – 1 8 × 1 4 ) +( 1 2 × – 1 4 × –1 ) ] –[ ( 1 2 × 1 2 × 1 4 ) +( 1 3 × – 1 8 × –1 ) +( 1 6 × – 1 4 × – 1 2 ) ] = [ – 1 6 +0– 1 2 ] –[ 1 8 +0– 1 12 ] [ – 1 12 – 1 192 + 1 8 ] –[ 1 16 + 1 24 + 1 48 ] = – 2 3 – 1 24 7 192 – 1 8 = – 17 24 – 17 = 8 z = | 1 3 – 1 4 1 1 6 1 2 1 1 2 – 1 8 0 | | 1 3 – 1 4 1 4 1 6 1 2 – 1 1 2 – 1 8 – 1 2 | = | 1 3 – 1 4 1 1 6 1 2 1 1 2 – 1 8 0 1 3 – 1 4 1 1 6 1 2 1 | | 1 3 – 1 4 1 4 1 6 1 2 – 1 1 2 – 1 8 – 1 2 1 3 – 1 4 1 4 1 6 1 2 – 1 | = [ ( 1 3 × 1 2 × 0 ) +( 1 6 × – 1 8 × 1 ) +( 1 2 × – 1 4 × 1 ) ] –[ ( 1 2 × 1 2 × 1 ) +( 1 3 × – 1 8 × 1 ) +( 1 6 × – 1 4 × 0 ) ] [ ( 1 3 × 1 2 × – 1 2 ) +( 1 6 × – 1 8 × 1 4 ) +( 1 2 × – 1 4 × –1 ) ] –[ ( 1 2 × 1 2 × 1 4 ) +( 1 3 × – 1 8 × –1 ) +( 1 6 × – 1 4 × – 1 2 ) ] = [ 0+– 1 48 – 1 8 ] –[ 1 4 – 1 24 +0 ] [ – 1 12 – 1 192 + 1 8 ] –[ 1 16 + 1 24 + 1 48 ] = – 7 48 – 5 24 7 192 – 1 8 = – 17 48 – 17 = 4 Sol.{ x=6 y=8 z=4
  12. { x 3 +y=2z+3 x–y=1 ↔ x+z= y 4 +11 { x 3 +y–2z=3 x–y=1 x– y 4 +z=11 x = | 3 1 – 2 1 – 1 0 11 – 1 4 1 | | 1 3 1 – 2 1 – 1 0 1 – 1 4 1 | = | 3 1 – 2 1 – 1 0 11 – 1 4 1 3 1 – 2 1 – 1 0 | | 1 3 1 – 2 1 – 1 0 1 – 1 4 1 1 3 1 – 2 1 – 1 0 | = [ ( 3 × –1 × 1 ) +( 1 × – 1 4 × –2 ) +( 11 × 1 × 0 ) ] –[ ( 11 × –1 × –2 ) +( 3 × – 1 4 × 0 ) +( 1 × 1 × 1 ) ] [ ( 1 3 × –1 × 1 ) +( 1 × – 1 4 × –2 ) +( 1 × 1 × 0 ) ] –[ ( 1 × –1 × –2 ) +( 1 3 × – 1 4 × 0 ) +( 1 × 1 × 1 ) ] = [ –3+ 1 2 +0 ] –[ 22+0+1 ] [ – 1 3 + 1 2 +0 ] –[ 2+0+1 ] = – 5 2 –23 1 6 –3 = – 2 – 17 = 9 y = | 1 3 3 – 2 1 1 0 1 11 1 | | 1 3 1 – 2 1 – 1 0 1 – 1 4 1 | = | 1 3 3 – 2 1 1 0 1 11 1 1 3 3 – 2 1 1 0 | | 1 3 1 – 2 1 – 1 0 1 – 1 4 1 1 3 1 – 2 1 – 1 0 | = [ ( 1 3 × 1 × 1 ) +( 1 × 11 × –2 ) +( 1 × 3 × 0 ) ] –[ ( 1 × 1 × –2 ) +( 1 3 × 11 × 0 ) +( 1 × 3 × 1 ) ] [ ( 1 3 × –1 × 1 ) +( 1 × – 1 4 × –2 ) +( 1 × 1 × 0 ) ] –[ ( 1 × –1 × –2 ) +( 1 3 × – 1 4 × 0 ) +( 1 × 1 × 1 ) ] = [ 1 3 –22+0 ] –[ –2+0+3 ] [ – 1 3 + 1 2 +0 ] –[ 2+0+1 ] = – 65 3 –1 1 6 –3 = – 3 – 17 = 8 z = | 1 3 1 3 1 – 1 1 1 – 1 4 11 | | 1 3 1 – 2 1 – 1 0 1 – 1 4 1 | = | 1 3 1 3 1 – 1 1 1 – 1 4 11 1 3 1 3 1 – 1 1 | | 1 3 1 – 2 1 – 1 0 1 – 1 4 1 1 3 1 – 2 1 – 1 0 | = [ ( 1 3 × –1 × 11 ) +( 1 × – 1 4 × 3 ) +( 1 × 1 × 1 ) ] –[ ( 1 × –1 × 3 ) +( 1 3 × – 1 4 × 1 ) +( 1 × 1 × 11 ) ] [ ( 1 3 × –1 × 1 ) +( 1 × – 1 4 × –2 ) +( 1 × 1 × 0 ) ] –[ ( 1 × –1 × –2 ) +( 1 3 × – 1 4 × 0 ) +( 1 × 1 × 1 ) ] = [ – 11 3 – 3 4 +1 ] –[ –3– 1 12 +11 ] [ – 1 3 + 1 2 +0 ] –[ 2+0+1 ] = – 41 12 – 95 12 1 6 –3 = – – 17 6 = 4 Sol.{ x=9 y=8 z=4
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