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1. El determinante de una matriz con una línea de ceros es cero.

3. El determinante de una matriz con una línea combinación
lineal de otras es cero.

2.  El determinante de una matriz con dos  líneas proporcionales es cero.

5. Si permutamos dos líneas, el determinante cambia de signo

 C_{1} ↔ C_{2} 

6. Si multiplicamos una línea por un número, el determinante 
queda multiplicado por ese número.

 \left| M \right| =\left| M^{t} \right| 

  \left| A·B \right| =\left| A  \right| ·\left| B \right| 

9. Suma de determinantes con todas las filas iguales excepto una.

10. El determinante de una matriz triangular es cero.

Cada sumando tiene un cero como factor.

Cada sumando tiene un cero como factor.

4. Si una línea se le suma una combinación lineal
de otras, el determinante no varía.

7. El determinante de una matriz y el de su traspuesta coinciden.

8. El determinante de un producto de matrices
es igual al producto de los determinantes.

\left|A  \right| ·\left|A^{-1}  \right| =1

\left|A^{-1}  \right| =\frac{1}{\left| A \right| }

Los productos de los elementos de una fila
por los adjuntos de otra.

El determinante de una matriz que ha sido 
multiplicada por un número.

Determinante de la matriz inversa

F34: 0
D42 = Random(0, 1)
D41 = If(D42 ≟ 0, Random[2, 4], (-Random[2, 4]))
A41 = Random(-6, 6)
C8 = Random(-9, 9)
G17 = Random(2, 4)
A17 = Random(-9, 9)
G11 = Random(2, 4)
B11 = Random(-9, 9)
A11 = Random(-9, 9)
C7 = Random(-9, 9)
A1 = Random(-9, 9)
B8 = Random(-9, 9)
B7 = Random(-9, 9)
E8 = Random(2, 4)
E7 = Random(2, 4)
A8 = Random(-9, 9)
D4 = Random(2, 4)
A7 = Random(-9, 9)
A4 = Random(-9, 9)
A2 = Random(-9, 9)
B1 = 0 (slider from -5 to 5, step 1)
B2: 0
B4 = Random(-9, 9)
A5: (A4 * D4)
B5: (B4 * D4)
A9: (A7 * E7) - (A8 * E8)
C9: (C7 * E7) - (C8 * E8)
B9: (B7 * E7) - (B8 * E8)
A12 = Random(-9, 9)
B12 = Random(-9, 9)
D12: A12 + (G11 * B12)
D11: A11 + (G11 * B11)
E11: B11
E12: B12
A14 = Random(-9, 9)
A15 = Random(-9, 9)
B14 = Random(-9, 9)
B15 = Random(-9, 9)
D14: B14
D15: B15
E14: A14
E15: A15
B17 = Random(-9, 9)
A18 = Random(-9, 9)
B18 = Random(-9, 9)
D18: A18
D17: (G17 * A17)
E17: (G17 * B17)
E18: B18
A20 = Random(-9, 9)
B20 = Random(-9, 9)
A21 = Random(-9, 9)
B21 = Random(-9, 9)
D20: A20
D21: B20
E20: A21
E21: B21
A23 = Random(-9, 9)
B23 = Random(-9, 9)
A24 = Random(-9, 9)
B24 = Random(-9, 9)
D23 = Random(-9, 9)
E23 = Random(-9, 9)
D24 = Random(-9, 9)
E24 = Random(-9, 9)
G23: (A23 * D23) + (B23 * D24)
H23: (A23 * E23) + (B23 * E24)
G24: (A24 * D23) + (B24 * D24)
H24: (A24 * E23) + (B24 * E24)
A26 = Random(-9, 9)
B26 = Random(-9, 9)
A27 = Random(-9, 9)
B27 = Random(-9, 9)
D26 = Random(-9, 9)
D27 = Random(-9, 9)
E26: B26
E27: B27
G26: A26 + D26
H26: B26
G27: A27 + D27
H27: B27
A29 = Random(-9, 9)
B29 = Random(-9, 9)
C29 = Random(-9, 9)
B30 = Random(-9, 9)
C30 = Random(-9, 9)
A30 = Random(-9, 9)
A31 = Random(-9, 9)
B31 = Random(-9, 9)
C31 = Random(-9, 9)
I1: {11, 12, 13}
i: SelectedIndex[I1] + 10
m1: {{A1, B1}, {A2, B2}}
m2: {{A4, B4}, {A5, B5}}
m3: {{A7, B7, C7}, {A8, B8, C8}, {A9, B9, C9}}
m41: {{A11, B11}, {A12, B12}}
m42: {{D11, E11}, {D12, E12}}
m51: {{A14, B14}, {A15, B15}}
m52: {{D14, E14}, {D15, E15}}
m61: {{A17, B17}, {A18, B18}}
m62: {{D17, E17}, {D18, E18}}
m71: {{A20, B20}, {A21, B21}}
m81: {{A23, B23}, {A24, B24}}
m82: {{D23, E23}, {D24, E24}}
m83: {{G23, H23}, {G24, H24}}
m91: {{A26, B26}, {A27, B27}}
m92: {{D26, E26}, {D27, E27}}
m93: {{G26, H26}, {G27, H27}}
ma1: {{A29, B29, C29}, {A30, B30, C30}, {A31, B31, C31}}
d1 = Determinant(m1)
texto1: "\left|" + (LaTeX[m1]) + "  \right| =\left( " + (LaTeX[A1]) + "  \right) ·\left(  " + (LaTeX[B2]) + " \right) -\left(" + (LaTeX[A2]) + "   \right)·\left(  " + (LaTeX[B1]) + " \right) =" + (LaTeX[d1]) + " "
d2 = Determinant(m2)
texto4: "\left| " + (LaTeX[m2]) + "  \right| =\left(   " + (LaTeX[A4]) + " \right) ·\left(  " + (LaTeX[B5]) + "  \right) -\left(  " + (LaTeX[A5]) + "  \right)·\left(  " + (LaTeX[B4]) + "  \right) =  " + (LaTeX[d2]) + ""
d3 = Determinant(m3)
d41 = Determinant(m41)
d42 = Determinant(m42)
d51 = Determinant(m51)
d52 = Determinant(m52)
d61 = Determinant(m61)
d62 = Determinant(m62)
d71 = Determinant(m71)
d81 = Determinant(m81)
d82 = Determinant(m82)
d83 = Determinant(m83)
d91 = Determinant(m91)
d92 = Determinant(m92)
d93 = Determinant(m93)
da1 = Determinant(ma1)
texto7: "\left|   " + (LaTeX[m3]) + "  \right| ="
texto6: "=\left(  " + (LaTeX[A7]) + "   \right) ·\left(  " + (LaTeX[B8]) + "   \right)  ·\left( " + (LaTeX[C9]) + "    \right) +\left(   " + (LaTeX[A8]) + "  \right) ·\left( " + (LaTeX[B9]) + "    \right)  ·\left(  " + (LaTeX[C7]) + "   \right) +\left(  " + (LaTeX[A9]) + "   \right) ·\left(  " + (LaTeX[C8]) + "   \right)  ·\left(  " + (LaTeX[B7]) + "   \right) -\left(   " + (LaTeX[A9]) + "  \right) ·\left(  " + (LaTeX[B8]) + "   \right)  ·\left( " + (LaTeX[C7]) + "    \right) -\left(  " + (LaTeX[B7]) + "  \right) ·\left(  " + (LaTeX[A8]) + " \right)  ·\left(   " + (LaTeX[C9]) + "  \right)  -\left( " + (LaTeX[A7]) + "    \right) ·\left(  " + (LaTeX[B9]) + "  \right)  ·\left(  " + (LaTeX[C8]) + "   \right)=" + (LaTeX[d3]) + ""
texto9: " F_{3}= " + (LaTeX[E7]) + "·F_{1}-" + (LaTeX[E8]) + "·F_{2}"
texto12: "\left| " + (LaTeX[m41]) + "  \right| =\left(  " + (LaTeX[A11]) + "  \right) ·\left(  " + (LaTeX[B12]) + "  \right) -\left(  " + (LaTeX[A12]) + "  \right)·\left( " + (LaTeX[B11]) + "   \right) =  " + (LaTeX[d41]) + ""
texto3: " F_{2} = " + (LaTeX[D4]) + " · F_{1}"
B = (-1, 2)
texto13: "\left|  " + (LaTeX[m42]) + " \right| =\left(  " + (LaTeX[D11]) + "  \right) ·\left(  " + (LaTeX[E12]) + "  \right) -\left(  " + (LaTeX[D12]) + "  \right)·\left( " + (LaTeX[E11]) + "   \right) = " + (LaTeX[d42]) + ""
texto16: "\left|  " + (LaTeX[m51]) + "  \right| =\left(  " + (LaTeX[A14]) + "   \right) ·\left(   " + (LaTeX[B15]) + "  \right) -\left(  " + (LaTeX[A15]) + "   \right)·\left(  " + (LaTeX[B14]) + "   \right) =   " + (LaTeX[d51]) + ""
texto17: "\left|  " + (LaTeX[m52]) + "  \right| =\left(   " + (LaTeX[D14]) + "  \right) ·\left(  " + (LaTeX[E15]) + "   \right) -\left(   " + (LaTeX[D15]) + "  \right)·\left( " + (LaTeX[E14]) + "    \right) =   " + (LaTeX[d52]) + ""
texto20: "\left|  " + (LaTeX[m61]) + "  \right| =\left(  " + (LaTeX[A17]) + "   \right) ·\left(   " + (LaTeX[B18]) + "  \right) -\left( " + (LaTeX[A18]) + "    \right)·\left(  " + (LaTeX[B17]) + "   \right) =  " + (LaTeX[d61]) + " "
texto21: "\left|  " + (LaTeX[m62]) + "  \right| =\left(  " + (LaTeX[D17]) + "   \right) ·\left(   " + (LaTeX[E18]) + "  \right) -\left(  " + (LaTeX[D18]) + "   \right)·\left(  " + (LaTeX[E17]) + "   \right) =   " + (LaTeX[d62]) + ""
m72: {{D20, E20}, {D21, E21}}
d72 = Determinant(m72)
texto26: "\left| A \right| =\left| " + (LaTeX[m81]) + "    \right| =\left(   " + (LaTeX[A23]) + "    \right) ·\left(    " + (LaTeX[B24]) + "   \right) -\left(  " + (LaTeX[A24]) + "     \right)·\left(   " + (LaTeX[B23]) + "    \right) =   " + (LaTeX[d81]) + "  "
texto27: "\left| B \right| =\left|  " + (LaTeX[m82]) + "    \right| =\left(  " + (LaTeX[D23]) + "      \right) ·\left(   " + (LaTeX[E24]) + "     \right) -\left(     " + (LaTeX[D24]) + "   \right)·\left(   " + (LaTeX[E23]) + "     \right) =    " + (LaTeX[d82]) + "  "
texto28: "\left| A·B \right| =\left|  " + (LaTeX[m83]) + "    \right| =\left(     " + (LaTeX[G23]) + "   \right) ·\left(   " + (LaTeX[H24]) + "     \right) -\left(    " + (LaTeX[G24]) + "    \right)·\left(  " + (LaTeX[H23]) + "      \right) =      " + (LaTeX[d83]) + ""
texto23: "\left|  M\right|= \left|  " + (LaTeX[m71]) + "  \right| =\left(  " + (LaTeX[A20]) + "    \right) ·\left(  " + (LaTeX[B21]) + "    \right) -\left(   " + (LaTeX[A21]) + "   \right)·\left(    " + (LaTeX[B20]) + "  \right) =    " + (LaTeX[d71]) + ""
texto24: "\left| M^{t} \right| =\left|  " + (LaTeX[m72]) + "   \right| =\left(  " + (LaTeX[D20]) + "    \right) ·\left(    " + (LaTeX[E21]) + "  \right) -\left(  " + (LaTeX[D21]) + "    \right)·\left(  " + (LaTeX[E20]) + "    \right) =    " + (LaTeX[d72]) + ""
texto30: "\left| " + (LaTeX[m91]) + "     \right| +\left|   " + (LaTeX[m92]) + "    \right| =\left|   " + (LaTeX[m93]) + "    \right| "
texto31: "\left|   " + (LaTeX[m91]) + "   \right| =\left( " + (LaTeX[A26]) + "       \right) ·\left(   " + (LaTeX[B27]) + "     \right) -\left(   " + (LaTeX[A27]) + "     \right)·\left(   " + (LaTeX[B26]) + "     \right) =      " + (LaTeX[d91]) + ""
texto32: "\left|   " + (LaTeX[m92]) + "   \right| =\left(  " + (LaTeX[D26]) + "   \right) ·\left(     " + (LaTeX[E27]) + "   \right) -\left(  " + (LaTeX[D27]) + "      \right)·\left(     " + (LaTeX[E26]) + "   \right) =     " + (LaTeX[d92]) + " "
texto33: "\left|  " + (LaTeX[m93]) + "    \right| =\left(   " + (LaTeX[G26]) + "     \right) ·\left(  " + (LaTeX[H27]) + "      \right) -\left(     " + (LaTeX[G27]) + "   \right)·\left(   " + (LaTeX[H26]) + "     \right) =      " + (LaTeX[d93]) + ""
texto35: "\left|   " + (LaTeX[ma1]) + "   \right| ="
texto36: "=\left(  " + (LaTeX[A29]) + "    \right) ·\left(  " + (LaTeX[B30]) + "    \right)  ·\left(   " + (LaTeX[C31]) + "   \right) +\left(  " + (LaTeX[A30]) + "    \right) ·\left(   " + (LaTeX[B31]) + "   \right)  ·\left(   " + (LaTeX[C29]) + "   \right) +\left(    " + (LaTeX[A31]) + "  \right) ·\left(  " + (LaTeX[C30]) + "    \right)  ·\left(  " + (LaTeX[B29]) + "    \right) -\left(  " + (LaTeX[A31]) + "    \right) ·\left(  " + (LaTeX[B30]) + "    \right)  ·\left(   " + (LaTeX[C29]) + "   \right) -\left(  " + (LaTeX[A29]) + "   \right) ·\left( " + (LaTeX[B31]) + "   \right)  ·\left(  " + (LaTeX[C30]) + "    \right)  -\left(   " + (LaTeX[B29]) + "   \right) ·\left(  " + (LaTeX[A30]) + "   \right)  ·\left(  " + (LaTeX[C31]) + "    \right)=" + (LaTeX[da1]) + " "
texto39: "Propiedad " + i + ""
texto19: " F_{1}^{'} = " + (LaTeX[G17]) + " · F_{1} "
texto11: " C_{1}^{'} = C_{1} + " + (LaTeX[G11]) + " · C_{2}"
A33 = Random(-2, 2)
B33 = Random(-2, 2)
C33 = Random(-2, 2)
A34 = Random(-2, 2)
B34 = Random(-2, 2)
C34 = Random(-2, 2)
A35 = Random(-2, 2)
B35 = Random(-2, 2)
C35 = Random(-2, 2)
mb3: {{A33, B33, C33}, {A34, B34, C34}, {A35, B35, C35}}
ld1: {-2, -1, 1, 2}
ld3: {0.2, 0.3, 0.3, 0.2}
E34 = RandomDiscrete(ld1, ld3)
E33 = RandomDiscrete(ld1, ld3)
F33 = RandomDiscrete(ld1, ld3)
G34 = RandomDiscrete(ld1, ld3)
G33 = RandomDiscrete(ld1, ld3)
E35 = RandomDiscrete(ld1, ld3)
F35 = RandomDiscrete(ld1, ld3)
G35 = RandomDiscrete(ld1, ld3)
mb4: {{E33, F33, G33}, {E34, F34, G34}, {E35, F35, G35}}
mb1 = If(Determinant[mb3] ≟ 0, mb4, mb3)
dmb3 = Determinant(mb3)
dmb4 = Determinant(mb4)
mb2: mb1^(-1)
texto42: "A= " + (LaTeX[mb1]) + ""
texto43: "A^{-1}=" + (LaTeX[mb2]) + ""
I33: 1
I34: 0
I35: 0
J35: 0
J34: 1
J33: 0
K33: 0
K34: 0
K35: 1
mbi: {{I33, J33, K33}, {I34, J34, K34}, {I35, J35, K35}}
texto44: "A·A^{-1}=" + (LaTeX[mb1]) + "·" + (LaTeX[mb2]) + "=" + (LaTeX[mbi]) + "=I"
A37 = Random(-6, 6)
B37 = Random(-6, 6)
C37 = Random(-6, 6)
A38 = Random(-6, 6)
B38 = Random(-6, 6)
C38 = Random(-6, 6)
A39 = Random(-6, 6)
B39 = Random(-6, 6)
C39 = Random(-6, 6)
B41 = Random(-6, 6)
A42 = Random(-6, 6)
B42 = Random(-6, 6)
E37: (A37 * $D$41)
E38: (A38 * $D$41)
E39: (A39 * $D$41)
F37: (B37 * $D$41)
F38: (B38 * $D$41)
F39: (B39 * $D$41)
G37: (C37 * $D$41)
G38: (C38 * $D$41)
G39: (C39 * $D$41)
mc1: {{A37, B37, C37}, {A38, B38, C38}, {A39, B39, C39}}
mc2: {{E37, F37, G37}, {E38, F38, G38}, {E39, F39, G39}}
mc3: {{A41, B41}, {A42, B42}}
dc1 = Determinant(mc1)
dc2 = Determinant(mc2)
dc3 = Determinant(mc3)
D40 = If(D42 ≟ 1, Random[2, 4], (-Random[2, 4]))
C41 = NaN
F41: (A41 * $D$40)
G41: (B41 * $D$40)
F42: (A42 * $D$40)
G42: (B42 * $D$40)
mc4: {{F41, G41}, {F42, G42}}
texto50: "B= " + (LaTeX[D40]) + "·A= " + (LaTeX[D40]) + "·" + (LaTeX[mc3]) + " = " + (LaTeX[mc4]) + ""
dc4 = Determinant(mc4)
texto52: "det(B)=\left( " + (LaTeX[D40]) + " \right)  ^{2}·\left( " + (LaTeX[dc3]) + " \right) = " + (LaTeX[dc4]) + ""
texto51: "det(A)= " + (LaTeX[dc3]) + ""
M29: B30
N29: C30
M30: B31
N30: C31
O29: A30
P29: C30
Q29: A30
R29: B30
O30: A31
P30: C31
Q30: A31
R30: B31
M31: B29
N31: C29
M32: B31
N32: C31
O31: A29
P31: C29
O32: A31
P32: C31
Q31: A29
Q32: A31
R31: B29
R32: B31
M33: B29
N33: C29
M34: B30
N34: C30
O33: A29
P33: C29
O34: A30
P34: C30
Q33: A29
R33: B29
Q34: A30
R34: B30
n11: {{M29, N29}, {M30, N30}}
n12: {{O29, P29}, {O30, P30}}
n13: {{Q29, R29}, {Q30, R30}}
n21: {{M31, N31}, {M32, N32}}
n22: {{O31, P31}, {O32, P32}}
n23: {{Q31, R31}, {Q32, R32}}
n31: {{M33, N33}, {M34, N34}}
n32: {{O33, P33}, {O34, P34}}
n33: {{Q33, R33}, {Q34, R34}}
matriz1: {{Q33, R33}, {Q34, R34}}
m = 2 (slider from 1 to 3, step 1)
n = 3 (slider from 1 to 3, step 1)
texto53: "Elementos de la fila " + m + ""
nn: {{A29, B29, C29}, {A30, B30, C30}, {A31, B31, C31}}
texto55: "a_{ " + (LaTeX[m]) + "1} Δ_{ " + (LaTeX[n]) + "1}+a_{  " + (LaTeX[m]) + "2} Δ_{ " + (LaTeX[n]) + "2}+a_{ " + (LaTeX[m]) + "3} Δ_{" + (LaTeX[n]) + "3}="
texto56: "A=" + (LaTeX[nn]) + ""
g1 = Element(nn, m, 1)
g2 = Element(nn, m, 2)
g3 = Element(nn, m, 3)
mg: {{M29, N29, O29, P29, Q29, R29}, {M30, N30, O30, P30, Q30, R30}, {M31, N31, O31, P31, Q31, R31}, {M32, N32, O32, P32, Q32, R32}, {M33, N33, O33, P33, Q33, R33}, {M34, N34, O34, P34, Q34, R34}}
p1 = Sequence(Sequence[Element[mg, i, j], j, 1, 2], i, (2 * n) - 1, (2 * n))
texto57: "" + (LaTeX[mg]) + ""
p2 = Sequence(Sequence[Element[mg, i, j], j, 3, 4], i, (2 * n) - 1, (2 * n))
p3 = Sequence(Sequence[Element[mg, i, j], j, 5, 6], i, (2 * n) - 1, (2 * n))
dp1: (Determinant[p1] * (-1)^(n + 1))
dp2: (Determinant[p2] * (-1)^(n + 2))
dp3: (Determinant[p3] * (-1)^(n + 3))
dq1 = Element(nn, m, 1)
dq2 = Element(nn, m, 2)
dq3 = Element(nn, m, 3)
dr1: (dp1 * dq1)
dr2: (dp2 * dq2)
dr3: (dp3 * dq3)
drr: dr1 + dr2 + dr3
texto59: "=\left( " + (LaTeX[dq1]) + " \right) ·\left( " + (LaTeX[dp1]) + "  \right) +\left(  " + (LaTeX[dq2]) + "\right) ·\left(  " + (LaTeX[dp2]) + " \right) +\left( " + (LaTeX[dq3]) + " \right) ·\left( " + (LaTeX[dp3]) + " \right) ="
texto60: "=\left(  " + (LaTeX[dr1]) + " \right) +\left(" + (LaTeX[dr2]) + "   \right) +\left(  " + (LaTeX[dr3]) + " \right) =" + (LaTeX[drr]) + ""
texto54: "Adjuntos de la fila " + n + ""
ds = Determinant(nn)
texto63: "A=" + (LaTeX[mc3]) + ""
texto64: "C=" + (LaTeX[mc1]) + ""
texto47: "D=" + (LaTeX[D41]) + "·C=" + (LaTeX[D41]) + "·" + (LaTeX[mc1]) + "=" + (LaTeX[mc2]) + ""
texto48: "det(C)=" + (LaTeX[dc1]) + ""
texto49: "det(D)=\left( " + (LaTeX[D41]) + " \right) ^{3}·\left( " + (LaTeX[dc1]) + " \right) =" + (LaTeX[dc2]) + ""
texto65: "A=" + (LaTeX[mb1]) + ""
texto58: "=\left( " + (LaTeX[dq1]) + "  \right) ·det" + (LaTeX[p1]) + "·\left(-1   \right) ^{" + (LaTeX[n]) + "+1}+\left(  " + (LaTeX[dq2]) + " \right) · det" + (LaTeX[p2]) + "·\left(-1   \right) ^{ " + (LaTeX[n]) + "+2}+\left(  " + (LaTeX[dq3]) + " \right) ·det" + (LaTeX[p3]) + "\left(-1   \right) ^{ " + (LaTeX[n]) + "+3}="