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Matemáticas 3 · Tema 6

Determinantes 3×3

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Ejemplo

{3x−2y+4z=2x+y−5z=72x−5y−z=5\begin{cases} 3x - 2y + 4z = 2 \\ x + y - 5z = 7 \\ 2x - 5y - z = 5 \end{cases}
Calculamos xx
x=∣2−2471−55−5−12−2471−5∣∣3−2411−52−5−13−2411−5∣x = \dfrac{\begin{vmatrix} 2 & -2 & 4 \\ 7 & 1 & -5 \\ 5 & -5 & -1 \\ 2 & -2 & 4 \\ 7 & 1 & -5 \end{vmatrix}}{\begin{vmatrix} 3 & -2 & 4 \\ 1 & 1 & -5 \\ 2 & -5 & -1 \\ 3 & -2 & 4 \\ 1 & 1 & -5 \end{vmatrix}}
=−2−140+50−(14+50+20)−3−20+20−(2+75+8)=−92−84−3−85=−176−88= \dfrac{-2 - 140 + 50 - (14 + 50 + 20)}{-3 - 20 + 20 - (2 + 75 + 8)} = \dfrac{-92 - 84}{-3 - 85} = \dfrac{-176}{-88}
x=2x = 2
Calculamos yy
y=∣32417−525−132417−5∣−88y = \dfrac{\begin{vmatrix} 3 & 2 & 4 \\ 1 & 7 & -5 \\ 2 & 5 & -1 \\ 3 & 2 & 4 \\ 1 & 7 & -5 \end{vmatrix}}{-88}
=−21+20−20−(−2−75+56)−88=−21+21−88=0−88= \dfrac{-21 + 20 - 20 - (-2 - 75 + 56)}{-88} = \dfrac{-21 + 21}{-88} = \dfrac{0}{-88}
y=0y = 0
Calculamos zz
z=∣3−221172−553−22117∣−88z = \dfrac{\begin{vmatrix} 3 & -2 & 2 \\ 1 & 1 & 7 \\ 2 & -5 & 5 \\ 3 & -2 & 2 \\ 1 & 1 & 7 \end{vmatrix}}{-88}
=15−10−28−(−10−105+4)−88=−23+111−88=88−88= \dfrac{15 - 10 - 28 - (-10 - 105 + 4)}{-88} = \dfrac{-23 + 111}{-88} = \dfrac{88}{-88}
z=−1z = -1

Comprobación

3(2)−2(0)+4(−1)=26−0−4=22=2\begin{aligned} 3(2) - 2(0) + 4(-1) &= 2 \\ 6 - 0 - 4 &= 2 \\ 2 &= 2 \end{aligned}
(2)+(0)−5(−1)=72+0+5=77=7\begin{aligned} (2) + (0) - 5(-1) &= 7 \\ 2 + 0 + 5 &= 7 \\ 7 &= 7 \end{aligned}
2(2)−5(0)−(−1)=54−0+1=55=5\begin{aligned} 2(2) - 5(0) - (-1) &= 5 \\ 4 - 0 + 1 &= 5 \\ 5 &= 5 \end{aligned}

Ejercicios

Ejercicio 1
{−x+2y+z=5−x+3y−2z=112x−y+z=−4\begin{cases} -x + 2y + z = 5 \\ -x + 3y - 2z = 11 \\ 2x - y + z = -4 \end{cases}
Ejercicio 2
{2x+3y−z=5x−y+z=13x+y−z=7\begin{cases} 2x + 3y - z = 5 \\ x - y + z = 1 \\ 3x + y - z = 7 \end{cases}
Ejercicio 3
{x−2y+3z=5−x+y−2z=−32x−3y−z=8\begin{cases} x - 2y + 3z = 5 \\ -x + y - 2z = -3 \\ 2x - 3y - z = 8 \end{cases}

Respuestas

▼
Ejercicio 1
{−x+2y+z=5−x+3y−2z=112x−y+z=−4\begin{cases} -x + 2y + z = 5 \\ -x + 3y - 2z = 11 \\ 2x - y + z = -4 \end{cases}
Calculamos xx
x=∣521113−2−4−11521113−2∣∣−121−13−22−11−121−13−2∣x = \dfrac{\begin{vmatrix} 5 & 2 & 1 \\ 11 & 3 & -2 \\ -4 & -1 & 1 \\ 5 & 2 & 1 \\ 11 & 3 & -2 \end{vmatrix}}{\begin{vmatrix} -1 & 2 & 1 \\ -1 & 3 & -2 \\ 2 & -1 & 1 \\ -1 & 2 & 1 \\ -1 & 3 & -2 \end{vmatrix}}
=15−11+16−(22+10−12)−3+1−8−(−2−2+6)=20−20−10−2=0−12= \dfrac{15 - 11 + 16 - (22 + 10 - 12)}{-3 + 1 - 8 - (-2 - 2 + 6)} = \dfrac{20 - 20}{-10 - 2} = \dfrac{0}{-12}
x=0x = 0
Calculamos yy
y=∣−151−111−22−41−151−111−2∣−12y = \dfrac{\begin{vmatrix} -1 & 5 & 1 \\ -1 & 11 & -2 \\ 2 & -4 & 1 \\ -1 & 5 & 1 \\ -1 & 11 & -2 \end{vmatrix}}{-12}
=−11+4−20−(−5−8+22)−12=−27−9−12=−36−12= \dfrac{-11 + 4 - 20 - (-5 - 8 + 22)}{-12} = \dfrac{-27 - 9}{-12} = \dfrac{-36}{-12}
y=3y = 3
Calculamos zz
z=∣−125−13112−1−4−125−1311∣−12z = \dfrac{\begin{vmatrix} -1 & 2 & 5 \\ -1 & 3 & 11 \\ 2 & -1 & -4 \\ -1 & 2 & 5 \\ -1 & 3 & 11 \end{vmatrix}}{-12}
=12+5+44−(8+11+30)−12=61−49−12=12−12= \dfrac{12 + 5 + 44 - (8 + 11 + 30)}{-12} = \dfrac{61 - 49}{-12} = \dfrac{12}{-12}
z=−1z = -1

Comprobación

−(0)+2(3)+(−1)=5−0+6−1=55=5\begin{aligned} -(0) + 2(3) + (-1) &= 5 \\ -0 + 6 - 1 &= 5 \\ 5 &= 5 \end{aligned}
−(0)+3(3)−2(−1)=11−0+9+2=1111=11\begin{aligned} -(0) + 3(3) - 2(-1) &= 11 \\ -0 + 9 + 2 &= 11 \\ 11 &= 11 \end{aligned}
2(0)−(3)+(−1)=−40−3−1=−4−4=−4\begin{aligned} 2(0) - (3) + (-1) &= -4 \\ 0 - 3 - 1 &= -4 \\ -4 &= -4 \end{aligned}
Ejercicio 2
{2x+3y−z=5x−y+z=13x+y−z=7\begin{cases} 2x + 3y - z = 5 \\ x - y + z = 1 \\ 3x + y - z = 7 \end{cases}
Calculamos xx
x=∣53−11−1171−153−11−11∣∣23−11−1131−123−11−11∣x = \dfrac{\begin{vmatrix} 5 & 3 & -1 \\ 1 & -1 & 1 \\ 7 & 1 & -1 \\ 5 & 3 & -1 \\ 1 & -1 & 1 \end{vmatrix}}{\begin{vmatrix} 2 & 3 & -1 \\ 1 & -1 & 1 \\ 3 & 1 & -1 \\ 2 & 3 & -1 \\ 1 & -1 & 1 \end{vmatrix}}
=5−1+21−(−3+5+7)2−1+9−(−3+2+3)=25−910−2=168= \dfrac{5 - 1 + 21 - (-3 + 5 + 7)}{2 - 1 + 9 - (-3 + 2 + 3)} = \dfrac{25 - 9}{10 - 2} = \dfrac{16}{8}
x=2x = 2
Calculamos yy
y=∣25−111137−125−1111∣8y = \dfrac{\begin{vmatrix} 2 & 5 & -1 \\ 1 & 1 & 1 \\ 3 & 7 & -1 \\ 2 & 5 & -1 \\ 1 & 1 & 1 \end{vmatrix}}{8}
=−2−7+15−(−5+14−3)8=6−68=08= \dfrac{-2 - 7 + 15 - (-5 + 14 - 3)}{8} = \dfrac{6 - 6}{8} = \dfrac{0}{8}
y=0y = 0
Calculamos zz
z=∣2351−113172351−11∣8z = \dfrac{\begin{vmatrix} 2 & 3 & 5 \\ 1 & -1 & 1 \\ 3 & 1 & 7 \\ 2 & 3 & 5 \\ 1 & -1 & 1 \end{vmatrix}}{8}
=−14+5+9−(21+2−15)8=0−88=−88= \dfrac{-14 + 5 + 9 - (21 + 2 - 15)}{8} = \dfrac{0 - 8}{8} = \dfrac{-8}{8}
z=−1z = -1

Comprobación

2(2)+3(0)−(−1)=54+0+1=55=5\begin{aligned} 2(2) + 3(0) - (-1) &= 5 \\ 4 + 0 + 1 &= 5 \\ 5 &= 5 \end{aligned}
(2)−(0)+(−1)=12−0−1=11=1\begin{aligned} (2) - (0) + (-1) &= 1 \\ 2 - 0 - 1 &= 1 \\ 1 &= 1 \end{aligned}
3(2)+(0)−(−1)=76+0+1=77=7\begin{aligned} 3(2) + (0) - (-1) &= 7 \\ 6 + 0 + 1 &= 7 \\ 7 &= 7 \end{aligned}
Ejercicio 3
{x−2y+3z=5−x+y−2z=−32x−3y−z=8\begin{cases} x - 2y + 3z = 5 \\ -x + y - 2z = -3 \\ 2x - 3y - z = 8 \end{cases}
Calculamos xx
x=∣5−23−31−28−3−15−23−31−2∣∣1−23−11−22−3−11−23−11−2∣x = \dfrac{\begin{vmatrix} 5 & -2 & 3 \\ -3 & 1 & -2 \\ 8 & -3 & -1 \\ 5 & -2 & 3 \\ -3 & 1 & -2 \end{vmatrix}}{\begin{vmatrix} 1 & -2 & 3 \\ -1 & 1 & -2 \\ 2 & -3 & -1 \\ 1 & -2 & 3 \\ -1 & 1 & -2 \end{vmatrix}}
=−5+27+32−(−6+30+24)−1+9+8−(−2+6+6)=54−4816−10=66= \dfrac{-5 + 27 + 32 - (-6 + 30 + 24)}{-1 + 9 + 8 - (-2 + 6 + 6)} = \dfrac{54 - 48}{16 - 10} = \dfrac{6}{6}
x=1x = 1
Calculamos yy
y=∣153−1−3−228−1153−1−3−2∣6y = \dfrac{\begin{vmatrix} 1 & 5 & 3 \\ -1 & -3 & -2 \\ 2 & 8 & -1 \\ 1 & 5 & 3 \\ -1 & -3 & -2 \end{vmatrix}}{6}
=3−24−20−(5−16−18)6=−41+296=−126= \dfrac{3 - 24 - 20 - (5 - 16 - 18)}{6} = \dfrac{-41 + 29}{6} = \dfrac{-12}{6}
y=−2y = -2
Calculamos zz
z=∣1−25−11−32−381−25−11−3∣6z = \dfrac{\begin{vmatrix} 1 & -2 & 5 \\ -1 & 1 & -3 \\ 2 & -3 & 8 \\ 1 & -2 & 5 \\ -1 & 1 & -3 \end{vmatrix}}{6}
=8+15+12−(16+9+10)6=35−356=06= \dfrac{8 + 15 + 12 - (16 + 9 + 10)}{6} = \dfrac{35 - 35}{6} = \dfrac{0}{6}
z=0z = 0

Comprobación

(1)−2(−2)+3(0)=51+4+0=55=5\begin{aligned} (1) - 2(-2) + 3(0) &= 5 \\ 1 + 4 + 0 &= 5 \\ 5 &= 5 \end{aligned}
−(1)+(−2)−2(0)=−3−1−2−0=−3−3=−3\begin{aligned} -(1) + (-2) - 2(0) &= -3 \\ -1 - 2 - 0 &= -3 \\ -3 &= -3 \end{aligned}
2(1)−3(−2)−(0)=82+6−0=88=8\begin{aligned} 2(1) - 3(-2) - (0) &= 8 \\ 2 + 6 - 0 &= 8 \\ 8 &= 8 \end{aligned}

Ejercicios extra

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Bloque 1
{−x−2y+z=63x−3y−3z=92x−3y−z=11\begin{cases} -x - 2y + z = 6 \\ 3x - 3y - 3z = 9 \\ 2x - 3y - z = 11 \end{cases}
{−2x−3y−3z=1x+y−3z=−8−2x−3y+2z=11\begin{cases} -2x - 3y - 3z = 1 \\ x + y - 3z = -8 \\ -2x - 3y + 2z = 11 \end{cases}
{x−y−z=−4−2x−2y+3z=1−2x−3y+2z=−2\begin{cases} x - y - z = -4 \\ -2x - 2y + 3z = 1 \\ -2x - 3y + 2z = -2 \end{cases}
{2x−y−z=−23x−y+2z=−8x+2y+z=−10\begin{cases} 2x - y - z = -2 \\ 3x - y + 2z = -8 \\ x + 2y + z = -10 \end{cases}
{−2x+3y−2z=6x+y+z=2−3x−2y+z=−8\begin{cases} -2x + 3y - 2z = 6 \\ x + y + z = 2 \\ -3x - 2y + z = -8 \end{cases}
Bloque 2
{−2x−2y−3z=0−2x−2y−2z=−23x−2y−3z=5\begin{cases} -2x - 2y - 3z = 0 \\ -2x - 2y - 2z = -2 \\ 3x - 2y - 3z = 5 \end{cases}
{−x−y−3z=−6−2x+y+2z=6−x+2y+2z=3\begin{cases} -x - y - 3z = -6 \\ -2x + y + 2z = 6 \\ -x + 2y + 2z = 3 \end{cases}
{−3x−3y+2z=19−2x+2y−3z=−4−x+2y−3z=−7\begin{cases} -3x - 3y + 2z = 19 \\ -2x + 2y - 3z = -4 \\ -x + 2y - 3z = -7 \end{cases}
{x−2y+3z=−2−x−y+2z=4−x+y−3z=1\begin{cases} x - 2y + 3z = -2 \\ -x - y + 2z = 4 \\ -x + y - 3z = 1 \end{cases}
{x+y−z=−3−3x−2y−3z=−83x−y+3z=5\begin{cases} x + y - z = -3 \\ -3x - 2y - 3z = -8 \\ 3x - y + 3z = 5 \end{cases}
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