Matrix Inverse with Gauss-Jordan Elimination

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2019-06-22 01:10

This GeoGebra example provides an interactive demonstration of computing a matrix inverse using Gauss-Jordan elimination. It visually shows how to transform an augmented matrix into reduced row echelon form through elementary row operations, thereby obtaining the inverse matrix. The core mathematical concepts include matrix inverse, elementary row operations, augmented matrices, and the Gauss-Jordan elimination algorithm. The construction utilizes GeoGebra commands such as Sequence, Flatten, Take, IndexOf, CountIf, Join, Remove, Insert, If, Sort, Element, Append, Textfield, TableText, and Expand, with objects including matrix elements a_1 through a_41, lists lista1, listaf1, listaf5, matrix matriz11, and variables n and m. This resource is suitable for high school and undergraduate linear algebra courses, helping students understand and verify the steps of matrix inversion.

GeoGebra Commands & Parameters

Sequence(Sequence[If[i ≟ j, 1, 0], i, 1, n], j, 1, n) → matriz11
Flatten(Append[Take[lista1, 1, m], Element[matriz11, 1]]) → listaf1
Flatten(Append[Take[lista7, 1, m], Element[matriz11, 5]]) → listaf5
Flatten(Append[Take[lista8, 1, m], Element[matriz11, 6]]) → listaf6
Flatten(Append[Take[lista2, 1, m], Element[matriz11, 2]]) → listaf2
Flatten(Append[Take[lista5, 1, m], Element[matriz11, 3]]) → listaf3
Take({a_1, a_8, a_{15}, a_{22}, a_{29}, a_{36}}, 1, n) → listac1
Flatten(Append[Take[lista6, 1, m], Element[matriz11, 4]]) → listaf4
IndexOf(f, listac1) → h
CountIf(t ≟ f, t, listac1) → e
Sequence((f(x) * i / i), i, 1, e) → lista4
Sequence(Take[lista4, 1, i], i, 1, e - 1) → lista9
Sequence(IndexOf[f, Remove[listac1, Element[lista9, i]]], i, 1, e - 1) → lista10
Sequence(Element[lista10, i] + i, i, 1, e - 1) → lista11
Join({{h}, lista11}) → lista13
Remove(Sequence[i, i, 1, n], lista13) → lista14
Remove(Sequence[i, i, 1, n], {Element[lista14, 1], Element[lista13, 1]}) → lista15
Insert(Element[lista14, 1], lista15, 1) → lista16
If(Element[listac1, 1] ≟ f, Insert[Element[lista13, 1], lista16, Element[lista14, 1]], Sequence[i, i, 1, n]) → lista17
Sort(Take[{listaf1, listaf2, listaf3, listaf4, listaf5, listaf6}, 1, n], lista17) → matriz1
Sequence(If[i ≟ 1, Sequence[Expand[Element[matriz1, i, j] / Element[matriz1, 1, 1]], j, 1, (2 * n)], Element[matriz1, i]], i, 1, n) → matriz2
Sequence(If[Element[matriz2, i, 1] ≠ f, Sequence[Expand[((-Element[matriz2, i, 1]) * Element[matriz2, 1, j]) + Element[matriz2, i, j]], j, 1, (2 * n)], Element[matriz1, i]], i, 2, n) → matriz3
Insert({Element[matriz2, 1]}, matriz3, 1) → lista18
If(Element[lista18, 2, 2] ≠ f, {1, 2, 3, 4}, If[Element[lista18, 3, 2] ≠ f, {1, 3, 2, 4}, {1, 4, 3, 2}]) → r_1
Sort(lista18, r_1) → matriz18
Element(lista18, 3, 2) → r
Element(listac1, Element[lista14, 1]) → p
Append(Sequence[Expand[Element[lista1, i] / a_1(x)], i, 1, m], Expand[Element[listater, 1] / a_1(x)]) → listaf12
Take({"R_1", "R_2", "R_3", "R_4", "R_5", "R_6"}, 1, n) → lista3
Element(lista3, Element[lista13, 1]) → texto38
Element(lista3, Element[lista14, 1]) → texto39
Textfield(a_{21}) → c_{13}
Textfield(a_{28}) → c_{14}
Textfield(a_{35}) → c_{15}
Textfield(a_{42}) → c_{16}
Textfield(a_{28}) → c_{24}
Textfield(a_{35}) → c_{25}
Textfield(a_{42}) → c_{26}
Textfield(a_{35}) → c_{35}
Textfield(a_{42}) → c_{36}
Textfield(a_{42}) → c_{46}
Textfield(a_1) → a_{73}
Textfield(a_2) → a_{74}
Textfield(a_3) → a_{77}
Textfield(a_4) → a_{47}
Textfield(a_5) → a_{75}
Textfield(a_6) → a_{48}
Textfield(a_8) → a_{49}
Textfield(a_9) → a_{76}
Textfield(a_{10}) → a_{67}
Textfield(a_{11}) → a_{68}
Textfield(a_{12}) → a_{69}
Textfield(a_{13}) → a_{70}
Textfield(a_{15}) → a_{50}
Textfield(a_{16}) → a_{57}
Textfield(a_{17}) → a_{58}
Textfield(a_{18}) → a_{59}
Textfield(a_{19}) → a_{60}
Textfield(a_{20}) → a_{78}
Textfield(a_{22}) → a_{71}
Textfield(a_{23}) → a_{72}
Textfield(a_{24}) → a_{43}
Textfield(a_{25}) → a_{44}
Textfield(a_{26}) → a_{45}
Textfield(a_{27}) → a_{46}
Textfield(a_{29}) → a_{51}
Textfield(a_{30}) → a_{52}
Textfield(a_{31}) → a_{53}
Textfield(a_{32}) → a_{54}
Textfield(a_{33}) → a_{55}
Textfield(a_{34}) → a_{56}
Textfield(a_{36}) → a_{61}
Textfield(a_{37}) → a_{62}
Textfield(a_{38}) → a_{63}
Textfield(a_{39}) → a_{64}
Textfield(a_{40}) → a_{65}
Textfield(a_{41}) → a_{66}
Take({a_2, a_9, a_{16}, a_{23}, a_{30}, a_{37}}, 1, n) → listac2
Take({a_3, a_{10}, a_{17}, a_{24}, a_{31}, a_{38}}, 1, n) → listac3
Take({a_4, a_{11}, a_{18}, a_{25}, a_{32}, a_{39}}, 1, n) → listac4
Take({a_5, a_{12}, a_{19}, a_{26}, a_{33}, a_{40}}, 1, n) → listac5
Take({a_6, a_{13}, a_{20}, a_{27}, a_{34}, a_{41}}, 1, n) → listac6
IndexOf(f, listac1) → g
Remove(listac1, {f, f}) → lista12
Element(lista3, 1) → texto41
If(Element[listac1, 1] ≟ f, texto40, texto43) → texto42
If(n ≟ 1, 0, If[n ≟ 2, 4, If[n ≟ 3, 7, If[n ≟ 4, 10, If[n ≟ 5, 10, 12]]]]) → b_1
If(m ≟ 6, 115, If[m ≟ 5, 120, If[m ≟ 4, 155, If[m ≟ 3, 195, If[m ≟ 2, 230]]]]) → w
Textfield(n) → CasillaDeEntrada2
TableText(Take[{listaf1, listaf2, listaf3, listaf4, listaf5, listaf6}, 1, n], If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → matriz1_1
TableText(matriz2, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto1
TableText(lista18, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto2
Sequence(If[Element[matriz2, i, 1] ≟ f, " ", (((Expand[(-Element[matriz2, i, 1])] * Element[lista3, 1]) * "+") * Element[lista3, i])], i, 2, n) → lista20
TableText({lista20}, "cv") → texto4
If(k > l ∧ k > 1, -1, If[k < l ∧ k ≥ 1, 1, 0]) → j
Sequence(If[i ≟ 2, Sequence[Expand[Element[lista18, i, j] / Element[lista18, 2, 2]], j, 1, (2 * n)], Element[lista18, i]], i, 1, n) → matriz5
Take({lista1, lista2, lista5, lista6, lista7, lista8}, 1, n) → matriz6
Sequence(Sequence[Element[matriz6, j, i], i, 1, n], j, 1, n) → matriz7
TableText(matriz5, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto3
If(n ≟ 2 ∧ k > 2, 60, If[n ≟ 3 ∧ k > 2, 285, If[n ≟ 4 ∧ k > 2, 408, 0]]) → a
Element(lista18, 2, 2) → q
Sequence(If[i ≟ 2, Element[matriz5, 2], Sequence[Expand[((-Element[matriz5, i, 2]) * Element[matriz5, 2, j]) + Element[matriz5, i, j]], j, 1, (2 * n)]], i, 1, n) → matriz8
TableText(matriz8, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto7
Sequence(If[Element[matriz2, i, 1] ≟ f, " ", (((Expand[(-Element[matriz2, i, 1])] * Element[lista3, 1]) * "+") * Element[lista3, i])], i, 2, n) → lista21
Sequence(If[Element[matriz5, i, 2] ≟ f, " ", If[i ≟ 2, " ", (((Expand[(-Element[matriz5, i, 2])] * Element[lista3, 2]) * "+") * Element[lista3, i])]], i, 1, n) → lista22
TableText({lista22}, "cv") → texto10
If(Element[listac1, 1] ≟ f, texto40, texto43) → texto13
TableText({lista20}, "cv") → texto15
TableText(matriz2, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto16
TableText(lista18, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto17
Sort(lista18, {1, 3, 2}) → matriz9
If(Element[lista18, 2, 2] ≟ f, texto21, texto19) → texto22
If(Element[lista18, 2, 2] ≟ f, " ", texto8) → texto26
Sort(lista18, {1, 3, 2}) → matriz12
Sequence(If[i ≟ 1, Sequence[Expand[Element[matriz1, i, j] / Element[matriz1, 1, 1]], j, 1, (2 * n)], Element[matriz1, i]], i, 1, n) → matriz13
Sequence(If[i ≟ 2, Sequence[Expand[Element[If[q ≟ f, matriz9, lista18], i, j] / Element[If[q ≟ f, matriz9, lista18], 2, 2]], j, 1, (2 * n)], Element[If[q ≟ f, matriz9, lista18], i]], i, 1, n) → matriz14
TableText(matriz14, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto25
Sequence(If[i ≟ 3, Sequence[Expand[(Element[matriz14, 2, j] * (-Element[matriz14, 3, 2])) + Element[matriz14, 3, j]], j, 1, (2 * n)], Element[matriz14, i]], i, 1, n) → matriz10
Expand((-Element[matriz14, 3, 2])) → s
TableText(matriz10, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto24
Element(matriz10, 3, 3) → t
Sequence(If[i ≟ 3, Sequence[Expand[Element[matriz10, 3, j] / If[Element[matriz10, 3, 3] ≟ f, 1, Element[matriz10, 3, 3]]], j, 1, (2 * n)], Element[matriz10, i]], i, 1, n) → matriz15
TableText(matriz15, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto30
Sequence(If[i ≠ 3, Sequence[Expand[((-Element[matriz15, i, 3]) * Element[matriz15, 3, j]) + Element[matriz15, i, j]], j, 1, (2 * n)], Element[matriz15, i]], i, 1, n) → matriz16
TableText(matriz16, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto33
Expand((-Element[matriz15, 2, 3])) → f_1
Expand((-Element[matriz15, 1, 3])) → p_1
Sequence(If[i ≟ 1, Sequence[Expand[((-Element[matriz16, 1, 2]) * Element[matriz16, 2, j]) + Element[matriz16, i, j]], j, 1, (2 * n)], Element[matriz16, i]], i, 1, n) → matriz17
Expand((-Element[matriz16, 1, 2])) → q_1
TableText(matriz17, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto37
If(Element[listac1, 1] ≟ f, texto40, texto43) → texto47
TableText(matriz2, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto49
TableText({lista20}, "cv") → texto52
TableText(lista18, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto53
IndexOf(r_1, Sequence[Element[lista18, i, 2], i, 2, 4]) → c
If(Element[lista18, 2, 2] ≠ f, {1, 2, 3, 4}, If[Element[lista18, 3, 2] ≠ f, {1, 3, 2, 4}, {1, 4, 3, 2}]) → lista23
Element(lista18, 4, 2) → s_1
TableText(matriz19, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto61
Element(matriz19, 3, 3) → t_1
Element(matriz19, 4, 3) → f_2
If(Element[lista18, 2, 2] ≠ f, texto54, If[Element[lista18, 3, 2] ≠ f, texto55, texto57]) → texto58
If(r ≟ f ∧ s ≟ f, " ", texto66) → texto65
TableText(matriz20, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto68
Element(matriz20, 3, 3) → p_2
Element(matriz20, 4, 3) → q_2
If(p_2 ≠ f, {1, 2, 3, 4}, {1, 2, 4, 3}) → lista24
Sort(matriz20, lista24) → matriz21
If(p_2 ≟ f, texto72, texto73) → texto74
TableText(matriz22, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto77
TableText(matriz23, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto83
Element(matriz23, 4, 4) → s_2
TableText(matriz24, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto87
TableText(matriz25, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto90
TableText(matriz26, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto94
TableText(matriz27, If[m ≟ 6, "c() |0000001000000", If[m ≟ 5, "c() |00000100000", If[m ≟ 4, "c() |000010000", If[m ≟ 3, "c() |0001000", If[m ≟ 2, "c() |00100", "c() |010"]]]]]) → texto97

AI Drawing Prompt Reference

Create an interactive diagram demonstrating matrix inversion via Gauss-Jordan elimination. Show an augmented matrix with the original square matrix on the left (elements as editable variables) and the identity matrix on the right. Display step-by-step row operations (row swapping, scaling, addition) using text labels and highlight the active rows. Use input fields for matrix entries so users can modify values and see the inverse update. Arrange the layout to clearly visualize the elimination process, with the final inverse matrix appearing on the right side.

The prompt above was compiled from the original drawing commands and is for reference only.