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Ch7Functions.py
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#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
"""
# Import the necessaries libraries
# Set notebook mode to work in offline
from __future__ import division
import numpy as np
from IPython.display import display, Latex, display_latex, Markdown
import plotly
import plotly.graph_objs as go
import matplotlib.pylab as plt
from matplotlib.patches import Polygon
from matplotlib.collections import PolyCollection
import plotly.offline as pyo
import plotly.graph_objs as go
import ipywidgets as widgets
from ipywidgets import interact, interactive, fixed, interact_manual, Layout, HBox, VBox
plotly.offline.init_notebook_mode(connected=True)
from IPython.core.magic import register_cell_magic
from IPython.display import HTML
import ipywidgets as widgets
import random
from ipywidgets import interact_manual, Layout
import sympy as sp
@register_cell_magic
def bgc(color):
script = (
"var cell = this.closest('.jp-CodeCell');"
"var editor = cell.querySelector('.jp-Editor');"
"editor.style.background='{}';"
"this.parentNode.removeChild(this)"
).format(color)
display(HTML('<img src onerror="{}">'.format(script)))
###############################################################################
## Chapter 7.1
def red_matrix(A, i, j):
""" Return reduced matrix (without row i and col j)"""
row = [0, 1, 2]
col = [0, 1, 2]
row.remove(i-1)
col.remove(j-1)
return A[row, col]
def pl_mi(i,j, first=False):
""" Return '+', '-' depending on row and col index"""
if (-1)**(i+j)>0:
if first:
return ""
else:
return "+"
else:
return "-"
def brackets(expr):
"""Takes a sympy expression, determine if it needs parenthesis and returns a string containing latex of expr
with or without the parenthesis."""
expr_latex = sp.latex(expr)
if '+' in expr_latex or '-' in expr_latex:
return "(" + expr_latex + ")"
else:
return expr_latex
def ch7ex(matrix):
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
Determinant_3x3(matrix, step_by_step=True)
button.on_click(solution)
display(box)
display(out)
def Determinant_3x3(A, step_by_step=True ,row=True, n=1):
"""
Step by step computation of the determinant of a 3x3 sympy matrix strating with given row/col number
:param A: 3 by 3 sympy matrix
:param step_by_step: Boolean, True: print step by step derivation of det, False: print only determinant
:param row: True to compute determinant from row n, False to compute determinant from col n
:param n: row or col number to compute the determinant from (int between 1 and 3)
:return: display step by step solution for
"""
if A.shape!=(3,3):
raise ValueError('Dimension of matrix A should be 3x3. The input A must be a sp.Matrix of shape (3,3).')
if n<1 or n>3 or not isinstance(n, int):
raise ValueError('n should be an integer between 1 and 3.')
# Construct string for determinant of matrix A
detA_s = sp.latex(A).replace('[','|').replace(']','|')
# To print all the steps
if step_by_step:
# If we compute the determinant with row n
if row:
# Matrix with row i and col j removed (red_matrix(A, i, j))
A1 = red_matrix(A, n, 1)
A2 = red_matrix(A, n, 2)
A3 = red_matrix(A, n, 3)
detA1_s = sp.latex(A1).replace('[','|').replace(']','|')
detA2_s = sp.latex(A2).replace('[','|').replace(']','|')
detA3_s = sp.latex(A3).replace('[','|').replace(']','|')
line1 = "$" + detA_s + ' = ' + pl_mi(n,1, True) + sp.latex(A[n-1, 0]) + detA1_s + pl_mi(n,2) + \
sp.latex(A[n-1, 1]) + detA2_s + pl_mi(n,3) + sp.latex(A[n-1, 2]) + detA3_s + '$'
line2 = '$' + detA_s + ' = ' + pl_mi(n,1, True) + sp.latex(A[n-1, 0]) + "\cdot (" + sp.latex(sp.det(A1)) \
+")" + pl_mi(n,2) + sp.latex(A[n-1, 1]) + "\cdot (" + sp.latex(sp.det(A2)) + ")"+ \
pl_mi(n,3) + sp.latex(A[n-1, 2]) + "\cdot (" + sp.latex(sp.det(A3)) + ')$'
line3 = '$' + detA_s + ' = ' + sp.latex(sp.simplify(sp.det(A))) + '$'
# If we compute the determinant with col n
else:
# Matrix with row i and col j removed (red_matrix(A, i, j))
A1 = red_matrix(A, 1, n)
A2 = red_matrix(A, 2, n)
A3 = red_matrix(A, 3, n)
detA1_s = sp.latex(A1).replace('[','|').replace(']','|')
detA2_s = sp.latex(A2).replace('[','|').replace(']','|')
detA3_s = sp.latex(A3).replace('[','|').replace(']','|')
line1 = "$" + detA_s + ' = ' + pl_mi(n,1, True) + brackets(A[0, n-1]) + detA1_s + pl_mi(n,2) + \
brackets(A[1, n-1]) + detA2_s + pl_mi(n,3) + brackets(A[2, n-1]) + detA3_s + '$'
line2 = '$' + detA_s + ' = ' + pl_mi(n,1, True) + brackets(A[0, n-1]) + "\cdot (" + sp.latex(sp.det(A1))\
+")" + pl_mi(n,2) + brackets(A[1, n-1]) + "\cdot (" + sp.latex(sp.det(A2)) + ")"+ \
pl_mi(n,3) + brackets(A[2, n-1]) + "\cdot (" + sp.latex(sp.det(A3)) + ')$'
line3 = '$' + detA_s + ' = ' + sp.latex(sp.simplify(sp.det(A))) + '$'
# Display step by step computation of determinant
display(Latex(line1))
display(Latex(line2))
display(Latex(line3))
# Only print the determinant without any step
else:
display(Latex("$" + detA_s + "=" + sp.latex(sp.det(A)) + "$"))
def ch7ex3(matrix):
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
Sarrus_3x3(matrix)
button.on_click(solution)
display(box)
display(out)
def Sarrus_3x3(A):
if A.shape!=(3,3):
raise ValueError('Dimension of matrix A should be 3x3. The input A must be a sp.Matrix of shape (3,3).')
# Construct string for determinant of matrix A
detA_s = sp.latex(A).replace('[','|').replace(']','|')
A11 = sp.latex(A[0,0]); A12 = sp.latex(A[0,1]); A13 = sp.latex(A[0,2])
A21 = sp.latex(A[1,0]); A22 = sp.latex(A[1,1]); A23 = sp.latex(A[1,2])
A31 = sp.latex(A[2,0]); A32 = sp.latex(A[2,1]); A33 = sp.latex(A[2,2])
x1 = '(' + A11 + '\cdot ' + A22 + '\cdot ' + A33 + ')'
x2 = '(' + A12 + '\cdot ' + A23 + '\cdot ' + A31 + ')'
x3 = '(' + A13 + '\cdot ' + A21 + '\cdot ' + A32 + ')'
y1 = '(' + A13 + '\cdot ' + A22 + '\cdot ' + A31 + ')'
y2 = '(' + A11 + '\cdot ' + A23 + '\cdot ' + A32 + ')'
y3 = '(' + A12 + '\cdot ' + A21 + '\cdot ' + A33 + ')'
prod1 = A[0,0] * A[1,1] * A[2,2];
prod2 = A[0,1] * A[1,2] * A[2,0];
prod3 = A[0,2] * A[1,0] * A[2,1];
prod4 = A[0,2] * A[1,1] * A[2,0];
prod5 = A[0,0] * A[1,2] * A[2,1];
prod6 = A[0,1] * A[1,0] * A[2,2];
def cplus(ex):
# Function to return the correct sign to print after a plus
if sp.simplify(ex) < 0:
string = " "
else:
string = "+"
return string
def cminus(ex):
# Function to return the correct sing to print after a plus
if sp.simplify(ex) < 0:
string = " "
else:
string = "-"
return string
display(Latex('$' + detA_s + ' = ' + x1 + "+" + x2 + "+" + x3 + "-" + y1 + "-" + y2 + "-" + y3 + '$' ))
display(Latex('$' + detA_s + ' = ' + sp.latex(sp.simplify(prod1)) + cplus(prod2) + sp.latex(sp.simplify(prod2)) + cplus(prod3) \
+ sp.latex(sp.simplify(prod3)) + " - [" + sp.latex(sp.simplify(prod4)) + cplus(prod5) + \
sp.latex(sp.simplify(prod5)) + cplus(prod6) + sp.latex(sp.simplify(prod6)) + '] $'))
display(Latex('$' + detA_s + ' = ' + sp.latex(sp.simplify(prod1+prod2+prod3)) + cminus(prod4+prod5 +prod6) \
+ sp.latex(sp.simplify(prod4+prod5+prod6)) + '$'))
display(Latex('$' + detA_s + ' = ' + sp.latex(sp.simplify(sp.det(A))) + '$'))
## 7.2
def question7_3(reponse):
A = sp.Matrix([[4,6,-3], [-1,2,3], [4, -5, 1]])
if np.abs(reponse - sp.det(A)) != 0:
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
Determinant_3x3(A, step_by_step = True)
button.on_click(solution)
display(box)
display(out)
else:
display("Bravo! Vous avez trouvé la réponse. Utilisez cette réponse pour les prochaines questions.")
def question7_3a(reponse):
a = sp.Matrix([[4, -1, 4], [6, 2, -5], [-3, 3, 1]])
if np.abs(reponse - sp.det(a)) != 0:
display("Dans ce cas, cette matrice est la transposée de A. Les déterminants sont égaux.")
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
Determinant_3x3(a, step_by_step = True)
button.on_click(solution)
display(box)
display(out)
else:
display("Bravo! Vous avez trouvé la réponse")
def question7_3b(reponse):
b = sp.Matrix([[4, 6, -3], [7, 14, -3], [4, -5, 1]])
if np.abs(reponse - sp.det(b)) != 0:
display("Dans ce cas, la première rangée a été multiplié par deux et elle a été ajouté à la deuxième rangée")
display(Latex("$ Soit: 2 \\times R_1 + R_2 \\rightarrow R_2 $"))
display("Le déterminant est le même")
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
Determinant_3x3(b, step_by_step = True)
button.on_click(solution)
display(box)
display(out)
else:
display("Bravo! Vous avez trouvé la réponse")
def question7_3c(reponse):
c = sp.Matrix([[4,6,-3], [-4,8,12], [-4, 5, -1]])
if np.abs(reponse - sp.det(c)) != 0:
display("Dans ce cas, la deuxième rangée a été multiplié par 4 et la troisième par -1")
display(Latex("$ Soit: 4 \\times R_2 \\rightarrow R_2, -1 \\times R_3 \\rightarrow R_3 $"))
display("Le déterminant est donc:")
display(Latex('$ det|c| = (-1) \cdot (4) \cdot det|A| = -4 \cdot 155 = - 620 $'))
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
Determinant_3x3(c, step_by_step = True)
button.on_click(solution)
display(box)
display(out)
else:
display("Bravo! Vous avez trouvé la réponse")
def question7_2d(reponse):
d = sp.Matrix([[4, 6, -3], [-1,2,3], [-8,10,-2]])
if np.abs(reponse - sp.det(d)) != 0:
display("Dans ce cas, la troisième rangée est multipliée par -2 ")
display(Latex("$ Soit: -2 \\times R_3 \\rightarrow R_3 $"))
display("Le déterminant est donc:")
display(Latex("$ det|d| = (-2) \cdot det|A| = -2 \cdot 155 = - 310 $"))
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
Determinant_3x3(d, step_by_step = True)
button.on_click(solution)
display(box)
display(out)
else:
display("Bravo! Vous avez trouvé la réponse.")
def question7_2e(reponse):
e = sp.Matrix([[-1, 2, 3], [4,6,-3], [0,-11,4]])
if np.abs(reponse - sp.det(e)) != 0:
display("Dans ce cas, deux fois la première rangée a été ajouté à la troisième rangée.")
display("Aussi les premières deux rangées sont échangées.")
display(Latex("$ Soit: 2 \\times R_1 + R_3 \\rightarrow R_3, R_1 \\leftrightarrow R_2 $"))
display("Le déterminant est donc:")
display(Latex("$ det|e| = (-1) \cdot det|A| = (-1) \cdot 155 = - 155 $"))
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(k):
out.clear_output()
Determinant_3x3(e, step_by_step = True)
button.on_click(solution)
display(box)
display(out)
else:
display("Bravo! Vous avez trouvé la réponse.")
def question7_2f(reponse):
f = sp.Matrix([[4, -5, 8], [6, 10, 1], [-3, 15, -2]])
if np.abs(reponse - sp.det(f)) != 0:
display("Dans ce cas, la deuxième rangée a été multiplié par cinq.")
display("Après ça, c''est la transposée de la matrice.")
display(Latex("$ Soit: 5 \\times R_2 \\rightarrow R_2, \: et \: transposée $"))
display("Le déterminant est donc:")
display(Latex("$ det|f| = 5 \cdot det|A|^T = 5 \cdot det|A| = 5 \cdot 155 = 755 $"))
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
Determinant_3x3(f, step_by_step = True)
button.on_click(solution)
display(box)
display(out)
else:
display("Bravo! Vous avez trouvé la réponse.")
## 7.3
def whether_invertibleA(A):
"""Judge whether the matrix C is invertible by calculating the determinant."""
A_RREF = A.rref()[0]
A_det = sp.Float(A.det(), 4)
detA_s = sp.latex(A)
detAr_s = sp.latex(A_RREF)
Ar_det = sp.Float(A_RREF.det(), 4)
flag = None
select = widgets.SelectMultiple(
options = ['La matrice est inversible', "La matrice n'est pas inversible"],
description = 'Réponse: ',
disabled = False,
layout = Layout(width='auto', height = 'auto')
)
button = widgets.Button(description = 'Vérifier', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def callback(e):
out.clear_output()
with out:
if len(select.value) <= 0:
pass
elif len(select.value) >1:
display(Markdown('Seulement une réponse est requise'))
elif sp.det(A) == 0:
if "La matrice n'est pas inversible" not in select.value:
display(Markdown("Faux"))
question3a(A)
else:
display(Markdown("Correct!"))
elif sp.det(A) != 0:
if 'La matrice est inversible' not in select.value:
display(Markdown("Faux"))
question3a(A)
else:
display(Markdown("Correct!"))
button.on_click(callback)
display(select)
display(box)
display(out)
def whether_invertibleB(A):
"""Judge whether the matrix C is invertible by calculating the determinant."""
A_RREF = A.rref()[0]
A_det = sp.Float(A.det(), 4)
detA_s = sp.latex(A)
detAr_s = sp.latex(A_RREF)
Ar_det = sp.Float(A_RREF.det(), 4)
flag = None
select = widgets.SelectMultiple(
options = ['La matrice est inversible', "La matrice n'est pas inversible"],
description = 'Réponse: ',
disabled = False,
layout = Layout(width='auto', height = 'auto')
)
button = widgets.Button(description = 'Vérifier', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def callback(e):
out.clear_output()
with out:
if len(select.value) <= 0:
pass
elif len(select.value) >1:
display(Markdown('Seulement une réponse est requise'))
elif sp.det(A) == 0:
if "La matrice n'est pas inversible" not in select.value:
display(Markdown("Faux"))
question3b(A)
else:
display(Markdown("Correct!"))
elif sp.det(A) != 0:
if 'La matrice est inversible' not in select.value:
display(Markdown("Faux"))
question3b(A)
else:
display(Markdown("Correct!"))
button.on_click(callback)
display(select)
display(box)
display(out)
def whether_invertibleC(A):
"""Judge whether the matrix C is invertible by calculating the determinant."""
A_RREF = A.rref()[0]
A_det = sp.Float(A.det(), 4)
detA_s = sp.latex(A)
detAr_s = sp.latex(A_RREF)
Ar_det = sp.Float(A_RREF.det(), 4)
flag = None
select = widgets.SelectMultiple(
options = ['La matrice est inversible', "La matrice n'est pas inversible"],
description = 'Réponse: ',
disabled = False,
layout = Layout(width='auto', height = 'auto')
)
button = widgets.Button(description = 'Vérifier', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def callback(e):
out.clear_output()
with out:
if len(select.value) <= 0:
pass
elif len(select.value) >1:
display(Markdown('Seulement une réponse est requise'))
elif sp.det(A) == 0:
if "La matrice n'est pas inversible" not in select.value:
display(Markdown("Faux"))
question3c(A)
else:
display(Markdown("Correct!"))
elif sp.det(A) != 0:
if 'La matrice est inversible' not in select.value:
display(Markdown("Faux"))
question3c(A)
else:
display(Markdown("Correct!"))
button.on_click(callback)
display(select)
display(box)
display(out)
def question3a(A):
A_RREF = A.rref()[0]
A_det = sp.Float(A.det(), 4)
detA_s = sp.latex(A)
detAr_s = sp.latex(A_RREF)
Ar_det = sp.Float(A_RREF.det(), 4)
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
display(Latex("$" + "\det A"+"="+ "\det" + detA_s + "=" + "k \cdot"+ "\det" + detAr_s+ "= k \cdot"+ "{}".format(Ar_det)+"=" + "{}".format(A_det)+"$" ))
display(Latex("Où $k$ est une constante qui n'est pas égale à zéro. "))
if sp.det(A) == 0:
display(Latex("$\det A$ est égal à zéro, donc la matrice $A$ est singulière." ))
else:
display(Latex("$\det A $ n'est pas égal à zéro, donc la matrice $A$ est inversible."))
button.on_click(solution)
display(box)
display(out)
def question3b(A):
A_RREF = A.rref()[0]
A_det = sp.Float(A.det(), 4)
detA_s = sp.latex(A)
detAr_s = sp.latex(A_RREF)
Ar_det = sp.Float(A_RREF.det(), 4)
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
display(Latex("$" + "\det B"+"="+ "\det" + detA_s + "=" + "k \cdot"+ "\det" + detAr_s+ "= k \cdot"+ "{}".format(Ar_det)+"=" + "{}".format(A_det)+"$" ))
display(Latex("Où $k$ est une constante qui n'est pas égale à zéro. "))
if sp.det(A) == 0:
display(Latex("$\det B$ est égal à zéro, donc la matrice $B$ est singulière." ))
else:
display(Latex("$\det B $ n'est pas égal à zéro, donc la matrice $B$ est inversible."))
button.on_click(solution)
display(box)
display(out)
def question3c(A):
A_RREF = A.rref()[0]
A_det = sp.Float(A.det(), 4)
detA_s = sp.latex(A)
detAr_s = sp.latex(A_RREF)
Ar_det = sp.Float(A_RREF.det(), 4)
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
display(Latex("$" + "\det C"+"="+ "\det" + detA_s + "=" + "k \cdot"+ "\det" + detAr_s+ "= k \cdot"+ "{}".format(Ar_det)+"=" + "{}".format(A_det)+"$" ))
display(Latex("Où $k$ est une constante qui n'est pas égale à zéro. "))
if sp.det(A) == 0:
display(Latex("$\det C$ est égal à zéro, donc la matrice $C$ est singulière." ))
else:
display(Latex("$\det C $ n'est pas égal à zéro, donc la matrice $C$ est inversible."))
button.on_click(solution)
display(box)
display(out)
## 7.4
def question7_4a_solution(reponse):
if np.abs(reponse - 6/119) > 0.01:
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
display(Markdown("Faux."))
A = sp.Matrix([[-3,2,7], [1,-1,4], [2,5,-6]]) # det = 119
B = sp.Matrix([[-4, 3, 0],[9, -4, -1],[-7, 1, 1]]) # det = 6
Determinant_3x3(A, step_by_step = True)
Determinant_3x3(B, step_by_step = True)
display(Latex("$$ \det C = \det B \det A^{-1} = \det B \left( \\frac{1}{\det A} \\right) = \\frac{\det B}{\det A} = \\frac{6}{119}$$"))
button.on_click(solution)
display(box)
display(out)
else:
display("Bravo! Vous avez trouvé la réponse")
def question7_4b_solution(detC, somme):
A = sp.Matrix([[1, 3, -6], [5, 2, 2], [-2, -4, 9]])
B = sp.Matrix([[5, 0, -2], [10, -3, 7], [-1, 4, 1]])
C = sp.Matrix([[6, 3, -8], [15, -1, 9],[-3, 0, 10]])
flag1 = False
flag2 = False
if detC != sp.det(C):
display(Markdown('Votre solution pour $\det(A+B) $ est fausse'))
flag1 = True
if somme != (sp.det(A) + sp.det(B)):
display(Markdown('Votre solution pour $\det A + \det B$ est fausse'))
flag1 = True
else:
display(Markdown("Correct! $\det(A+B)$ n'est pas égale à $\det A + \det B $"))
if flag1:
question7_4b(A,B,C)
def question7_4b(A,B,C):
display(Markdown('Cliquez en sous pour la solution pour $\det(A+B)$ et pour $\det A + \det B $ '))
button1 = widgets.Button(description = 'Solution', disabled = False)
box1 = HBox(children = [button1])
out1 = widgets.Output()
@out1.capture()
def solution(e):
out1.clear_output()
display(Markdown('La solution pour $\det(A+B)$'))
Determinant_3x3(C, step_by_step = True)
display(Markdown('La solution pour $\det A$'))
Determinant_3x3(A, step_by_step = True)
display(Markdown('La solution pour $\det B$'))
Determinant_3x3(B, step_by_step = True)
button1.on_click(solution)
display(box1)
display(out1)
## 7.5
def plotDeterminant3D(A):
"""
This function is used to plot a 3D representation of a determinant as the volume
of a parallelopiped. This gets the vertices and sends it to another function below to plot.
"""
# Will only execute if it is 3x3
if (np.shape(A) != (3,3)):
print('La matrice A doit être 3x3.')
return
# This creates the parallelopiped coordonates
cube = np.array([[-1, -1, -1],
[1, -1, -1 ],
[1, 1, -1],
[-1, 1, -1],
[-1, -1, 1],
[1, -1, 1 ],
[1, 1, 1],
[-1, 1, 1]])
vertices = np.zeros((8,3))
for i in range(8):
vertices[i,:] = np.dot(cube[i,:], A)
data_x = vertices[:,0]
data_y = vertices[:,1]
data_z = vertices[:,2]
vol = np.abs(np.linalg.det(A))
vol = np.round(vol, decimals = 3)
plot3d(data_x, data_y, data_z, vol)
def plot3d(data_x, data_y, data_z, vol):
"""
This function plots an interactive 3D plot of the determinant as a volume of a
parallelopiped
"""
fig = go.Figure(
data = [
go.Mesh3d(
x = data_x,
y = data_y,
z = data_z,
i = [7, 0, 0, 0, 4, 4, 6, 6, 4, 0, 3, 2], # These are needed, numbers from documentation
j = [3, 4, 1, 2, 5, 6, 5, 2, 0, 1, 6, 3],
k = [0, 7, 2, 3, 6, 7, 1, 1, 5, 5, 7, 6],
colorscale=[[0, 'darkblue'],
[0.5, 'lightskyblue'],
[1, 'darkblue']],
intensity = np.linspace(0, 1, 8, endpoint=True),
showscale=False,
opacity = 0.6
)
],
layout = go.Layout(
title = "Le volume est: " + str(vol),
autosize = True
)
)
# This prints it
pyo.iplot(fig, filename='Determinant-Volume')
def plotDeterminant2D(A):
# See; https://stackoverflow.com/questions/44881885/python-draw-parallelepiped
"""
This function creates a 2D plot of the area of a determinant for a 2x2 matrix.
"""
# Will only execute if it is 2x2
if (np.shape(A) != (2,2)):
print('La matrice A doit être 2x2.')
return
# Define vertices for a cube to multiply with input matrix A to get parallelopiped
rect = np.array([[1, -1],
[1, 1],
[-1, 1],
[-1, -1]])
vertices = np.zeros((5,2))
for i in range(4): vertices[i,:] = np.dot(rect[i,:], A)
vertices[4,0] = vertices[0,0]
vertices[4,1] = vertices[0,1]
# Create figure / grid to plot
fig = plt.figure(figsize=(10,5))
ax = fig.add_subplot(111)
# Plot vertices
ax.plot(vertices[:,0], vertices[:,1])
vol = np.abs(np.linalg.det(A)) # absolute value of the determinant
vol = np.round(vol, decimals = 3)
print("L'aire est:", vol)
coll = PolyCollection([vertices])
ax.add_collection(coll)
ax.autoscale_view()
plt.grid()
plt.show()
def problem7_5():
A = [[1, 3], [5,2]]
display(Markdown('Cliquez en sous pour la solution pour votre dessin'))
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
plotDeterminant2D(A)
button.on_click(solution)
display(box)
display(out)
def Determinant_3x3_abs(A, step_by_step=True ,row=True, n=1):
"""
Step by step computation of the determinant of a 3x3 sympy matrix strating with given row/col number
:param A: 3 by 3 sympy matrix
:param step_by_step: Boolean, True: print step by step derivation of det, False: print only determinant
:param row: True to compute determinant from row n, False to compute determinant from col n
:param n: row or col number to compute the determinant from (int between 1 and 3)
:return: display step by step solution for
Same idea as for 7.1 but returns the absolute value of the result
"""
if A.shape!=(3,3):
raise ValueError('Dimension of matrix A should be 3x3. The input A must be a sp.Matrix of shape (3,3).')
if n<1 or n>3 or not isinstance(n, int):
raise ValueError('n should be an integer between 1 and 3.')
# Construct string for determinant of matrix A
detA_s = sp.latex(A).replace('[','|').replace(']','|')
# To print all the steps
if step_by_step:
# If we compute the determinant with row n
if row:
# Matrix with row i and col j removed (red_matrix(A, i, j))
A1 = red_matrix(A, n, 1)
A2 = red_matrix(A, n, 2)
A3 = red_matrix(A, n, 3)
detA1_s = sp.latex(A1).replace('[','|').replace(']','|')
detA2_s = sp.latex(A2).replace('[','|').replace(']','|')
detA3_s = sp.latex(A3).replace('[','|').replace(']','|')
line1 = "$" + detA_s + ' = ' + pl_mi(n,1, True) + sp.latex(A[n-1, 0]) + detA1_s + pl_mi(n,2) + \
sp.latex(A[n-1, 1]) + detA2_s + pl_mi(n,3) + sp.latex(A[n-1, 2]) + detA3_s + '$'
line2 = '$' + detA_s + ' = ' + pl_mi(n,1, True) + sp.latex(A[n-1, 0]) + "\cdot (" + sp.latex(sp.det(A1)) \
+")" + pl_mi(n,2) + sp.latex(A[n-1, 1]) + "\cdot (" + sp.latex(sp.det(A2)) + ")"+ \
pl_mi(n,3) + sp.latex(A[n-1, 2]) + "\cdot (" + sp.latex(sp.det(A3)) + ')$'
line3 = '$' + detA_s + ' = ' + sp.latex(sp.simplify(sp.det(A))) + '$'
# If we compute the determinant with col n
else:
# Matrix with row i and col j removed (red_matrix(A, i, j))
A1 = red_matrix(A, 1, n)
A2 = red_matrix(A, 2, n)
A3 = red_matrix(A, 3, n)
detA1_s = sp.latex(A1).replace('[','|').replace(']','|')
detA2_s = sp.latex(A2).replace('[','|').replace(']','|')
detA3_s = sp.latex(A3).replace('[','|').replace(']','|')
line1 = "$" + detA_s + ' = ' + pl_mi(n,1, True) + brackets(A[0, n-1]) + detA1_s + pl_mi(n,2) + \
brackets(A[1, n-1]) + detA2_s + pl_mi(n,3) + brackets(A[2, n-1]) + detA3_s + '$'
line2 = '$' + detA_s + ' = ' + pl_mi(n,1, True) + brackets(A[0, n-1]) + "\cdot (" + sp.latex(sp.det(A1))\
+")" + pl_mi(n,2) + brackets(A[1, n-1]) + "\cdot (" + sp.latex(sp.det(A2)) + ")"+ \
pl_mi(n,3) + brackets(A[2, n-1]) + "\cdot (" + sp.latex(sp.det(A3)) + ')$'
line3 = '$' + detA_s + ' = ' + sp.latex(sp.simplify(sp.Abs(sp.det(A)))) + '$'
# Display step by step computation of determinant
display(Latex(line1))
display(Latex(line2))
display(Latex(line3))
if sp.det(A) < 0:
display(Latex('$' + ' Le \: déterminant \: est \: négatif \: alors \: on \: prend \: la \: valeur \: absolue \: $'))
display(Latex("$ Le \: volume \: est \: = " + sp.latex(-1 * sp.det(A)) + "$"))
# Only print the determinant without any step
else:
if sp.det(A) > 0:
display(Latex("$" + detA_s + "=" + sp.latex(sp.det(A)) + "$"))
else:
display(Latex('$' + ' Le \: déterminant \: est \: négatif \: alors \: on \: prend \: la \: valeur \: absolue \: $'))
display(Latex("$ Le \: volume \: est \: = " + sp.latex(-1 * sp.det(A)) + "$"))
## 7.7
def find_inverse_3x3(A):
"""
Step by step computation of the determinant of a 3x3 sympy matrix strating with given row/col number
:param A: 3 by 3 sympy matrix
:param step_by_step: Boolean, True: print step by step derivation of det, False: print only determinant
:param row: True to compute determinant from row n, False to compute determinant from col n
:param n: row or col number to compute the determinant from (int between 1 and 3)
:return: display step by step solution for
"""
if A.shape!=(3,3):
raise ValueError(' La Matrice A doit être 3x3.')
if A.det() == 0:
display(Latex("$\det A=0.$" +" La matrice $A$ est singulière alors l'inverse de $A$ n'existe pas." ))
else:
sub_matrix=[]
pl=[]
cofactor=[]
cofactor_o=[]
sub_matrix_latex=[]
cof_matrix =sp.Matrix([[1,1,1],[1,1,1],[1,1,1]])
# Construc string for determinant of matrix A
detA_s = sp.latex(A).replace('[','|').replace(']','|')
for i in range(3):
for j in range(3):
sub_matrix.append(red_matrix(A, i+1, j+1))
pl.append( (-1)**(i+j+2))
for i in range(len(pl)):
cofactor.append(sp.latex(pl[i]*sp.simplify(sp.det(sub_matrix[i]))))
cofactor_o.append(pl[i]*sp.simplify(sp.det(sub_matrix[i])))
sub_matrix_latex.append(sp.latex(sub_matrix[i]).replace('[','|').replace(']','|'))
for i in range(3):
for j in range(3):
cof_matrix[i,j]=cofactor_o[i*3+j]
cof_matrix_t=cof_matrix.T
A1 = red_matrix(A, 1, 1)
A2 = red_matrix(A, 1, 2)
A3 = red_matrix(A, 1, 3)
detA1_s = sp.latex(A1).replace('[','|').replace(']','|')
detA2_s = sp.latex(A2).replace('[','|').replace(']','|')
detA3_s = sp.latex(A3).replace('[','|').replace(']','|')
c12 = sp.simplify(sp.det(A2))*(-1)
line0 = "$\mathbf{Solution:}$ Les neuf cofacteurs sont"
line1 = "$" + "C_{11}" + ' = ' + pl_mi(1,1, True) + sub_matrix_latex[0] + ' = ' + cofactor[0] + "\qquad " +\
"C_{12}" + ' = ' + pl_mi(1,2) + sub_matrix_latex[1] + ' = ' + cofactor[1]+ "\qquad " +\
"C_{13}" + ' = ' + pl_mi(1,3) + sub_matrix_latex[2] + ' = ' + cofactor[2]+ '$'
line2 = "$" + "C_{21}" + ' = ' + pl_mi(2,1, True) + sub_matrix_latex[3] + ' = ' + cofactor[3] + "\qquad " +\
"C_{22}" + ' = ' + pl_mi(2,2) + sub_matrix_latex[4] + ' = ' + cofactor[4]+ "\qquad " +\
"C_{23}" + ' = ' + pl_mi(2,3) + sub_matrix_latex[5] + ' = ' + cofactor[5]+ '$'
line3 = "$" + "C_{31}" + ' = ' + pl_mi(3,1, True) + sub_matrix_latex[6] + ' = ' + cofactor[6] + "\qquad " +\
"C_{32}" + ' = ' + pl_mi(3,2) + sub_matrix_latex[7] + ' = ' + cofactor[7]+ "\qquad " +\
"C_{33}" + ' = ' + pl_mi(3,3) + sub_matrix_latex[8] + ' = ' + cofactor[8]+ '$'
line4 = "La comatrice de $A$ est la transposée de la matrice de cofaceurs de $A$"
#"Then we can get the adjugate matrix that is the transpose of the matrix of cofactors. For instance, $C_{13}$ goes \
# in the $(3,1)$ position of the adjugate matrix."
line5 = '$'+"\mbox{adj} A" + ' = ' + "(\mbox{cof} A)^T" + ' = ' + sp.latex(cof_matrix_t) + '$'
line6 = "On calcule le déterminant de $A$: $det\ A$"
line7 = "$ \mbox{det} A=" + detA_s + "=" + sp.latex(sp.simplify(sp.det(A))) + '$'
line8 = "On peut trouver la matrice inverse en utilisant le théorème en haut."
line9 = '$'+"A^{-1}" + ' = ' + "\dfrac{{1}}{{\mbox{det} A}}"+"\cdot "+" \mbox{adj} A" + ' = ' \
+"\dfrac{1}"+"{{{}}}".format(sp.simplify(sp.det(A)))+"\cdot "+sp.latex(cof_matrix_t)+"="+ sp.latex(sp.simplify(sp.det(A))**(-1)*cof_matrix_t) + '$'
# Display step by step computation of determinant
display(Markdown('Cliquez en sous pour la solution'))
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
display(Latex(line0))
display(Latex(line1))
display(Latex(line2))
display(Latex(line3))
display(Latex(line4))
display(Latex(line5))
display(Latex(line6))
display(Latex(line7))
display(Latex(line8))
display(Latex(line9))
button.on_click(solution)
display(box)
display(out)
def red_matrix4(A, i, j):
""" Return reduced matrix (without row i and col j)"""
row = [0, 1, 2, 3]
col = [0, 1, 2, 3]
row.remove(i-1)
col.remove(j-1)
return A[row, col]
def find_inverse_4x4(A):
"""
Step by step computation of the determinant of a 4x4 sympy matrix strating with given row/col number
:param A: 3 by 3 sympy matrix
:param step_by_step: Boolean, True: print step by step derivation of det, False: print only determinant
:param row: True to compute determinant from row n, False to compute determinant from col n
:param n: row or col number to compute the determinant from (int between 1 and 3)
:return: display step by step solution for
"""
if A.shape!=(4,4):
raise ValueError(' La Matrice B doit être 4x4.')
if A.det() == 0:
display(Latex("$\det B=0.$" +" La matrice $B$ est singulière alors l'inverse de $B$ n'existe pas." ))
else:
sub_matrix=[]
pl=[]
cofactor=[]
cofactor_o=[]
sub_matrix_latex=[]
cof_matrix =sp.Matrix([[1,1,1,1],[1,1,1,1],[1,1,1,1],[1,1,1,1]])
# Construc string for determinant of matrix A
detA_s = sp.latex(A).replace('[','|').replace(']','|')
for i in range(4):
for j in range(4):
sub_matrix.append(red_matrix4(A, i+1, j+1))
pl.append( (-1)**(i+j+2))
for i in range(len(pl)):
cofactor.append(sp.latex(pl[i]*sp.simplify(sp.det(sub_matrix[i]))))
cofactor_o.append(pl[i]*sp.simplify(sp.det(sub_matrix[i])))
sub_matrix_latex.append(sp.latex(sub_matrix[i]).replace('[','|').replace(']','|'))
for i in range(4):
for j in range(4):
cof_matrix[i,j]=cofactor_o[i*4+j]
cof_matrix_t=cof_matrix.T
A1 = red_matrix4(A, 1, 1)
A2 = red_matrix4(A, 1, 2)
A3 = red_matrix4(A, 1, 3)
detA1_s = sp.latex(A1).replace('[','|').replace(']','|')
detA2_s = sp.latex(A2).replace('[','|').replace(']','|')
detA3_s = sp.latex(A3).replace('[','|').replace(']','|')
c12 = sp.simplify(sp.det(A2))*(-1)
line0 = "$\mathbf{Solution:}$ Les 16 cofacteurs sont"
line1 = "$" + "C_{11}" + ' = ' + pl_mi(1,1, True) + sub_matrix_latex[0] + ' = ' + cofactor[0] + "\qquad " +\
"C_{12}" + ' = ' + pl_mi(1,2) + sub_matrix_latex[1] + ' = ' + cofactor[1] + "\qquad " +\
"C_{13}" + ' = ' + pl_mi(1,3) + sub_matrix_latex[2] + ' = ' + cofactor[2] + "\qquad " +\
"C_{14}" + ' = ' + pl_mi(1,4) + sub_matrix_latex[3] + ' = ' + cofactor[3] + "\qquad " + '$'
line2 = "$" + "C_{21}" + ' = ' + pl_mi(2,1, True) + sub_matrix_latex[4] + ' = ' + cofactor[4] + "\qquad " +\
"C_{22}" + ' = ' + pl_mi(2,2) + sub_matrix_latex[5] + ' = ' + cofactor[5]+ "\qquad " +\
"C_{23}" + ' = ' + pl_mi(2,3) + sub_matrix_latex[6] + ' = ' + cofactor[6]+ "\qquad " + \
"C_{24}" + ' = ' + pl_mi(2,4) + sub_matrix_latex[7] + ' = ' + cofactor[7]+ "\qquad" + '$'
line3 = "$" + "C_{31}" + ' = ' + pl_mi(3,1, True) + sub_matrix_latex[8] + ' = ' + cofactor[8] + "\qquad " +\
"C_{32}" + ' = ' + pl_mi(3,2) + sub_matrix_latex[9] + ' = ' + cofactor[9] + "\qquad " + \
"C_{33}" + ' = ' + pl_mi(3,3) + sub_matrix_latex[10] + ' = ' + cofactor[10]+ "\qquad " + \
"C_{34}" + ' = ' + pl_mi(3,4) + sub_matrix_latex[11] + ' = ' + cofactor[11]+ "\qquad" +'$'
line4 = "$" + "C_{41}" + ' = ' + pl_mi(4,1, True) + sub_matrix_latex[12] + ' = ' + cofactor[12] + "\qquad " +\
"C_{42}" + ' = ' + pl_mi(4,2) + sub_matrix_latex[13] + ' = ' + cofactor[13] + "\qquad " +\
"C_{43}" + ' = ' + pl_mi(4,3) + sub_matrix_latex[14] + ' = ' + cofactor[14]+ "\qquad " +\
"C_{44}" + ' = ' + pl_mi(4,4) + sub_matrix_latex[15] + ' = ' + cofactor[15]+ "\qquad" +'$'
line5 = "Cela donne la matrice de cofacteurs de $B$."
line6 = '$'+ "(\mbox{cof} B)" + ' = ' + sp.latex(cof_matrix) + '$'
line7 = "La comatrice de $B$ est la transposée de la matrice de cofacteurs de $B$"
#"Then we can get the adjugate matrix that is the transpose of the matrix of cofactors. For instance, $C_{13}$ goes \
# in the $(3,1)$ position of the adjugate matrix."
line8 = '$'+"\mbox{adj} B" + ' = ' + "(\mbox{cof} B)^T" + ' = ' + sp.latex(cof_matrix_t) + '$'
line9 = "On calcule le déterminant de $B$: $det\ B$"
line10 = "$ \mbox{det} B=" + detA_s + "=" + sp.latex(sp.simplify(sp.det(A))) + '$'
line11 = "On peut trouver la matrice inverse en utilisant le théorème en haut."
line12 = '$'+"B^{-1}" + ' = ' + "\dfrac{{1}}{{\mbox{det} B}}"+"\cdot "+" \mbox{adj} B" + ' = ' \
+"\dfrac{1}"+"{{{}}}".format(sp.simplify(sp.det(A)))+"\cdot "+sp.latex(cof_matrix_t)+"="+ sp.latex(sp.simplify(sp.det(A))**(-1)*cof_matrix_t) + '$'
# Display step by step computation of determinant
display(Markdown('Cliquez en sous pour la solution'))
button = widgets.Button(description = 'Solution', disabled = False)
box = HBox(children = [button])
out = widgets.Output()
@out.capture()
def solution(e):
out.clear_output()
display(Latex(line0))
display(Latex(line1))
display(Latex(line2))
display(Latex(line3))
display(Latex(line4))
display(Latex(line5))
display(Latex(line6))
display(Latex(line7))
display(Latex(line8))
display(Latex(line9))
display(Latex(line10))
display(Latex(line11))
display(Latex(line12))
button.on_click(solution)
display(box)
display(out)

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