a_{12}

a_{13}

a_{21}

a_{22}

a_{23}

a_{31}

a_{32}

a_{33}

b1

b2

a_{11}

(  ) x + (  ) + (  ) z = 
(  ) x + (  ) + (  ) z = 
(  ) x + (  ) + (  ) z = 

¿Primera fila?

b3

Ceros \;  en \;  1^{a} \;  columna

SI

NO

NO

SI

Volver a comenzar

SI

NO

SI

NO

¿Cambia orden de filas?

SI

NO

SI

NO

SI

NO

SI

NO

Coeficientes del sistema

Ecuaciones

¿ a_{22}≟ 0?

Hacer \; a_{11}=1:

SI

NO

SI

NO

Hacer \; a_{22}=1:

Si  a_{22}= 0 Intercambia F_{2} con F_{3}

 Si a_{22}≠0 ir al siguiente paso

Intercambia F_{2} con F_{3}

¿ a_{33}≟ 0?

SI a_{33}=0 la matriz no tiene inversa

SI a_{33}≠0 ir al siguiente paso

La matriz no tiene inversa

Hacer \; a_{33}=1:

Ceros \;  en \;  2^{a} \;  columna

Ceros \;  en \;  3^{a} \;  columna

Método de Gauss-Jordan para
calcular la inversa de una matriz

SI

SI

NO

NO

La matriz y la Identidad

Hacer   a_{11} =1

Ceros en primera columna

Intercambia fila 2 con fila 3

Hacer  a_{22} =1

Ceros en segunda columna

Hacer   a_{33} =1

Ceros en la tercera columna

La matriz inversa

Si a_{11} = 0 intercambia filas

La matriz no tiene inversa

La matriz no tiene inversa

F1 = 0 (slider from -5 to 5, step 1)
F2 = 0
A1 = 4
A2 = 2
F6 = If($A$1 ≟ 0 ∧ ($A$2 ≠ 0), F1, F2)
F3 = 1
A3 = 5
F7 = If($A$1 ≟ 0 ∧ $A$2 ≟ 0 ∧ ($A$3 ≠ 0), F1, F3)
F5 = If($A$1 ≠ 0, F1, If[$A$2 ≠ 0, F2, F3])
E1 = 0
E2 = 1
E6 = If($A$1 ≟ 0 ∧ ($A$2 ≠ 0), E1, E2)
E3 = 0
E7 = If($A$1 ≟ 0 ∧ $A$2 ≟ 0 ∧ ($A$3 ≠ 0), E1, E3)
E5 = If($A$1 ≠ 0, E1, If[$A$2 ≠ 0, E2, E3])
D1 = 1
D2 = 0
D6 = If($A$1 ≟ 0 ∧ ($A$2 ≠ 0), D1, D2)
D3 = 0
D7 = If($A$1 ≟ 0 ∧ $A$2 ≟ 0 ∧ ($A$3 ≠ 0), D1, D3)
D5 = If($A$1 ≠ 0, D1, If[$A$2 ≠ 0, D2, D3])
C1 = 3
C2 = 5
C6 = If($A$1 ≟ 0 ∧ ($A$2 ≠ 0), C1, C2)
C3 = 3
C7 = If($A$1 ≟ 0 ∧ $A$2 ≟ 0 ∧ ($A$3 ≠ 0), C1, C3)
C5 = If($A$1 ≠ 0, C1, If[$A$2 ≠ 0, C2, C3])
B1 = 3
B2 = 4
B6 = If($A$1 ≟ 0 ∧ ($A$2 ≠ 0), B1, B2)
B3 = 3
B7 = If($A$1 ≟ 0 ∧ $A$2 ≟ 0 ∧ ($A$3 ≠ 0), B1, B3)
B5 = If($A$1 ≠ 0, B1, If[$A$2 ≠ 0, B2, B3])
A7_1 = NaN
A7 = If($A$1 ≟ 0 ∧ $A$2 ≟ 0 ∧ ($A$3 ≠ 0), A1, A3)
A6_1 = NaN
A6 = If($A$1 ≟ 0 ∧ ($A$2 ≠ 0), A1, A2)
A5_1 = NaN
A5 = If($A$1 ≠ 0, A1, If[$A$2 ≠ 0, A2, A3])
n11 = 0 (slider from 0 to 1, step 1)
n = 1 (slider from 1 to 3, step 1)
F9: F5
A9: A5
E9: E5
D9: D5
C9: C5
B9: B5
F14 = If(n11 ≟ 0, F9, F9 / $A9)
F18: F14
F22: F18
F10: F6
F15: F10
B11: B7
F11: F7
F16: F11
B10: B6
B16: B11
A11: A7
B15: B10
A10: A6
A16: A11
A15: A10
F20: F16 - (F14 * $A$16)
F19: F15 - (F14 * $A$15)
B14 = If(n11 ≟ 0, B9, B9 / $A9)
B20: B16 - (B14 * $A$16)
B19: B15 - (B14 * $A$15)
F23 = If($B$19 ≟ 0, F20, F19)
B23 = If($B$19 ≟ 0, B20, B19)
F24 = If($B$19 ≟ 0, F19, F20)
E14 = If(n11 ≟ 0, E9, E9 / $A9)
E18: E14
E22: E18
E10: E6
E15: E10
E11: E7
E16: E11
E20: E16 - (E14 * $A$16)
E19: E15 - (E14 * $A$15)
E23 = If($B$19 ≟ 0, E20, E19)
E24 = If($B$19 ≟ 0, E19, E20)
D14 = If(n11 ≟ 0, D9, D9 / $A9)
D18: D14
D10: D6
D22: D18
D11: D7
D15: D10
D16: D11
D20: D16 - (D14 * $A$16)
D19: D15 - (D14 * $A$15)
D23 = If($B$19 ≟ 0, D20, D19)
D24 = If($B$19 ≟ 0, D19, D20)
C14 = If(n11 ≟ 0, C9, C9 / $A9)
C18: C14
C10: C6
C22: C18
C11: C7
C15: C10
C16: C11
C20: C16 - (C14 * $A$16)
C19: C15 - (C14 * $A$15)
C23 = If($B$19 ≟ 0, C20, C19)
C24 = If($B$19 ≟ 0, C19, C20)
B18: B14
B22: B18
A14 = If(n11 ≟ 0, A9, A9 / $A9)
B24 = If($B$19 ≟ 0, B19, B20)
A19: A15 - (A14 * $A$15)
A20: A16 - (A14 * $A$16)
A24 = If($B$19 ≟ 0, A19, A20)
A23 = If($B$19 ≟ 0, A20, A19)
A18: A14
A22: A18
B26: B22
B28: B24
B27: B23 / $B$23
C28: C24
D28: D24
E28: E24
F28: F24
A28: A24
C27: C23 / $B$23
D27: D23 / $B$23
E27: E23 / $B$23
F27: F23 / $B$23
A27: A23 / $B$23
C26: C22
D26: D22
E26: E22
F26: F22
A26: A22
B21 = 3
C21 = 7
B30: B26 - (B27 * $B$26)
A30: A26 - (A27 * $B$26)
C30: C26 - (C27 * $B$26)
D30: D26 - (D27 * $B$26)
F30: F26 - (F27 * $B$26)
E30: E26 - (E27 * $B$26)
A12 = 1
B12 = 2
C31: C27
B31: B27
D31: D27
A31: A27
F31: F27
E31: E27
C12 = 1
C32: C28 - (C27 * $B$28)
D32: D28 - (D27 * $B$28)
D12 = 3
A32: A28 - (A27 * $B$28)
B32: B28 - (B27 * $B$28)
F32: F28 - (F27 * $B$28)
E32: E28 - (E27 * $B$28)
CasillaDeEntrada12 = Textfield(B1)
CasillaDeEntrada13 = Textfield(C1)
CasillaDeEntrada21 = Textfield(A2)
CasillaDeEntrada22 = Textfield(B2)
CasillaDeEntrada23 = Textfield(C2)
CasillaDeEntrada31 = Textfield(A3)
CasillaDeEntrada32 = Textfield(B3)
CasillaDeEntrada33 = Textfield(C3)
CasillaDeEntradab1 = Textfield(D1)
CasillaDeEntradab2 = Textfield(D2)
CasillaDeEntrada11 = Textfield(A1)
texto5b: "( " + A18 + " ) x + ( " + B18 + " ) y + ( " + C18 + " ) z = " + D18 + "
( " + A19 + " ) x + ( " + B19 + " ) y + ( " + C19 + " ) z = " + D19 + "
( " + A20 + " ) x + ( " + B20 + " ) y + ( " + C20 + " ) z = " + D20 + ""
texto5: "( " + A1 + " ) x + ( " + B1 + " ) y + ( " + C1 + " ) z = " + D1 + "
( " + A2 + " ) x + ( " + B2 + " ) y + ( " + C2 + " ) z = " + D2 + "
( " + A3 + " ) x + ( " + B3 + " ) y + ( " + C3 + " ) z = " + D3 + ""
texto5c: "( " + A30 + " ) x + ( " + B30 + " ) y + ( " + C30 + " ) z = " + D30 + "
( " + A31 + " ) x + ( " + B31 + " ) y + ( " + C31 + " ) z = " + D31 + "
( " + A32 + " ) x + ( " + B32 + " ) y + ( " + C32 + " ) z = " + D32 + ""
CasillaDeEntradat5 = Textfield(B21)
CasillaDeEntradat6 = Textfield(C21)
A44: D32 / C32
A43: (D31 - (A44 * C31)) / B31
A42: (D30 - (C30 * A44) - (B30 * A43)) / A30
texto5a: "( " + (LaTeX[A9]) + " ) x + ( " + (LaTeX[B9]) + " ) y + ( " + (LaTeX[C9]) + " ) z = " + (LaTeX[D9]) + "
( " + (LaTeX[A10]) + " ) x + ( " + (LaTeX[B10]) + " ) y + ( " + (LaTeX[C10]) + " ) z = " + (LaTeX[D10]) + "
( " + (LaTeX[A11]) + " ) x + ( " + (LaTeX[B11]) + " ) y + ( " + (LaTeX[C11]) + " ) z = " + (LaTeX[D11]) + ""
CasillaDeEntradab3 = Textfield(D1)
i = 1 (slider from 0 to 1, step 1)
j = 1 (slider from 0 to 1, step 1)
k = 1 (slider from 0 to 1, step 1)
n1 = 0 (slider from 0 to 1, step 1)
n2 = 0 (slider from 0 to 1, step 1)
n3 = 0 (slider from 0 to 1, step 1)
n4 = 0 (slider from 0 to 1, step 1)
A: {{A1, B1, C1, D1, E1, F1}, {A2, B2, C2, D2, E2, F2}, {A3, B3, C3, D3, E3, F3}}
B: {{A9, B9, C9, D9, E9, F9}, {A10, B10, C10, D10, E10, F10}, {A11, B11, C11, D11, E11, F11}}
D: {{A18, B18, C18, D18, E18, F18}, {A19, B19, C19, D19, E19, F19}, {A20, B20, C20, D20, E20, F20}}
G: {{A30, B30, C30, D30, E30, F30}, {A31, B31, C31, D31, E31, F31}, {A32, B32, C32, D32, E32, F32}}
texto2: "A=" + (LaTeX[A]) + ""
texto5e: "( " + (LaTeX[B31]) + " ) y + ( " + (LaTeX[C31]) + " ) · ( " + (LaTeX[A44]) + " ) = " + (LaTeX[D31]) + " \;  \;  ;\;  \;  y = " + (LaTeX[A43]) + ""
texto5f: "( " + (LaTeX[A30]) + " ) x + ( " + (LaTeX[B30]) + " ) · ( " + (LaTeX[A43]) + " ) + ( " + (LaTeX[C30]) + " ) · ( " + (LaTeX[A44]) + " ) = " + (LaTeX[D30]) + " \;  \; ;\;  \; x = " + (LaTeX[A42]) + ""
E12 = 3
F12 = 3
C: {{A14, B14, C14, D14, E14, F14}, {A15, B15, C15, D15, E15, F15}, {A16, B16, C16, D16, E16, F16}}
texto2b: "" + (LaTeX[C]) + ""
textoa = If(B19 ≟ 0, textoa1, textoa2)
F: {{A26, B26, C26, D26, E26, F26}, {A27, B27, C27, D27, E27, F27}, {A28, B28, C28, D28, E28, F28}}
texto3a: "F_{2}=F_{2}-(" + (LaTeX[A15]) + ")·F_{1}"
texto3b: "F_{3}=F_{3}-(" + (LaTeX[A16]) + ") ·F_{1}"
texto3g: "F_{3}=F_{3}-(" + (LaTeX[B28]) + " ) ·F_{2}"
texto3f: "F_{1}=F_{1}-(" + (LaTeX[B26]) + ") ·F_{2}"
textob = If(C32 ≟ 0, textob1, textob2)
A34: A30
A36: A32 / $C$32
A35: A31
B34: B30
B36: B32 / $C$32
B35: B31
C34: C30
C36: C32 / $C$32
C35: C31
D34: D30
D36: D32 / $C$32
D35: D31
E34: E30
E36: E32 / $C$32
E35: E31
F34: F30
F36: F32 / $C$32
F35: F31
texto3j: "F_{2}=F_{2}-( " + (LaTeX[C35]) + " ) ·F_{3}"
texto3i: "F_{1}=F_{1}-( " + (LaTeX[C34]) + ") ·F_{3}"
A40: A36
B40: B36
C40: C36
D40: D36
E40: E36
F40: F36
G36 = NaN
G35 = NaN
G34 = NaN
H36 = NaN
H35 = NaN
H34 = NaN
A38: A34 - (A36 * $C$34)
B38: B34 - (B36 * $C$34)
C38: C34 - (C36 * $C$34)
D38: D34 - (D36 * $C$34)
E38: E34 - (E36 * $C$34)
F38: F34 - (F36 * $C$34)
A39: A35 - (A36 * $C$35)
B39: B35 - (B36 * $C$35)
C39: C35 - (C36 * $C$35)
D39: D35 - (D36 * $C$35)
E39: E35 - (E36 * $C$35)
F39: F35 - (F36 * $C$35)
H: {{A34, B34, C34, D34, E34, F34}, {A35, B35, C35, D35, E35, F35}, {A36, B36, C36, D36, E36, F36}}
I: {{A38, B38, C38, D38, E38, F38}, {A39, B39, C39, D39, E39, F39}, {A40, B40, C40, D40, E40, F40}}
J: {{D38, E38, F38}, {D39, E39, F39}, {D40, E40, F40}}
texto1: "" + (LaTeX[B]) + ""
texto3: "" + (LaTeX[D]) + ""
texto6: "" + (LaTeX[F]) + ""
texto7: "" + (LaTeX[G]) + ""
texto8: "" + (LaTeX[H]) + ""
texto9: "" + (LaTeX[I]) + ""
texto10: "A^{-1}=" + (LaTeX[J]) + ""
E_1 = (-3.9687691435053885, 1.7463981849120793)
K = (-3.9488031764038567, -1.5479863868406147)
n5 = 0 (slider from 0 to 1, step 1)
E: {{A22, B22, C22, D22, E22, F22}, {A23, B23, C23, D23, E23, F23}, {A24, B24, C24, D24, E24, F24}}
lista1: {A5, B5, C5, D5, E5, F5, A6, B6, C6, D6, E6, F6, A7, B7, C7, D7, E7, F7}
AA_1: {{A5, B5, C5, D5, E5, F5}, {A6, B6, C6, D6, E6, F6}, {A7, B7, C7, D7, E7, F7}}
AA: {{A1, B1, C1}, {A2, B2, C2}, {A3, B3, C3}}
rg = MatrixRank(AA)
texto23: "A =  " + (LaTeX[A]) + " ≈ " + (LaTeX[AA_1]) + ""
texto11: "" + (LaTeX[D]) + "≈" + (LaTeX[E]) + ""
texto26: "\frac{F_{1} }{" + (LaTeX[A5]) + " }"
texto28: "\frac{F_{2}}{" + (LaTeX[B23]) + "}"
texto29: "\frac{F_{2}}{ " + (LaTeX[C32]) + "}"
texto25: "Para  \; que  \;  a_{11} =1 \; Hacemos\frac{F_{1} }{" + (LaTeX[A5]) + "}"