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#PUT CURSOR ANYWHERE IN THIS BOX THEN HIT SHIFT+ENTER to start the applet
@interact
def _(row = slider([1..10], label='rows', default = 3) , col = slider([1..10], label='columns', default = 3)):

    @interact(layout=dict(top=[['M', 'b']]))

    def gauss_method(b= random_matrix(QQ,nrows = row, ncols = 1, algorithm='echelonizable', rank = 1), M=random_matrix(QQ,nrows = row, ncols = col, algorithm='echelonizable', rank = min(row,col)),rescale_leading_entry=True ):
        """Describe the reduction to echelon form of the given matrix of rationals.

        M  matrix of rationals   e.g., M = matrix(QQ, [[..], [..], ..])
        rescale_leading_entry=False  boolean  make the leading entries to 1's

        Returns: None.  Side effect: M is reduced.  Note: this is echelon form,
        not reduced echelon form; this routine does not end the same way as does
        M.echelon_form().

        """

        M=M.augment(b, subdivide = True)
        N=copy(M)
        N=N.rref()


        num_rows=M.nrows()
        num_cols=M.ncols()
        print("Reduced echelon form:")
        show(N)

        print("Steps")
        show(M)



        col = 0   # all cols before this are already done
        for row in range(0,num_rows):
            # ?Need to swap in a nonzero entry from below
            while (col < num_cols
                   and M[row][col] == 0):
                for i in M.nonzero_positions_in_column(col):
                    if i > row:
                        print " R",row+1," swap R",i+1
                        M.swap_rows(row,i)
                        show(M)
                        break
                else:
                    col += 1

            if col >= num_cols:
                break

            # Now guaranteed M[row][col] != 0
            if (rescale_leading_entry
               and M[row][col] != 1):
                print "",1/M[row][col],"R",row+1
                M.rescale_row(row,1/M[row][col])
                show(M)
            change_flag=False
            C=1
            for changed_row in range(row+1,num_rows):
                if M[changed_row][col] != 0:
                    change_flag=True
                    factor=-1*M[changed_row][col]/M[row][col]
                    print "",factor," R",row+1,"+R",changed_row+1
                    M.add_multiple_of_row(changed_row,row,factor)
                C +=1
            if change_flag:
                show(M)
            col +=1

        print "Above is the echelon form, let's keep cruising to get the reduced echelon form"

        for changed_row in range(1,num_rows):
            for row in range(0,changed_row):
                factor = -1*M[row][changed_row]
                if factor != 0:
                    print "",factor," R",row+1,"+R",changed_row+1
                    M.add_multiple_of_row(row,changed_row,factor)
                    show(M)
Interact: please open in CoCalc
#Example with variables in entries

a= var('a')
b= var('b')
c= var('c')
h= var('h')
M = matrix(SR,[ [-1,4,1], [h,-2,-7], [7,5,2] ])
B= matrix(SR,[[a],[b],[c]])
M=M.augment(B,subdivide = True)
print "Starting M"
show(M)
print ""
print "Reduced M"
show(M.echelon_form())
Starting M
(−141ah−2−7b752c)\displaystyle \left(\begin{array}{rrr|r} -1 & 4 & 1 & a \\ h & -2 & -7 & b \\ 7 & 5 & 2 & c \end{array}\right)
Reduced M
(100−a+(14 a+2 c−33 (ah+b)2 h−1)(2 (h−7)2 h−1−1)3 (11 (h−7)2 h−1−6)+2 (ah+b)2 h−1010ah+b2 (2 h−1)+(14 a+2 c−33 (ah+b)2 h−1)(h−7)6 (2 h−1)(11 (h−7)2 h−1−6)001−14 a+2 c−33 (ah+b)2 h−13 (11 (h−7)2 h−1−6))\displaystyle \left(\begin{array}{rrr|r} 1 & 0 & 0 & -a + \frac{{\left(14 \, a + 2 \, c - \frac{33 \, {\left(a h + b\right)}}{2 \, h - 1}\right)} {\left(\frac{2 \, {\left(h - 7\right)}}{2 \, h - 1} - 1\right)}}{3 \, {\left(\frac{11 \, {\left(h - 7\right)}}{2 \, h - 1} - 6\right)}} + \frac{2 \, {\left(a h + b\right)}}{2 \, h - 1} \\ 0 & 1 & 0 & \frac{a h + b}{2 \, {\left(2 \, h - 1\right)}} + \frac{{\left(14 \, a + 2 \, c - \frac{33 \, {\left(a h + b\right)}}{2 \, h - 1}\right)} {\left(h - 7\right)}}{6 \, {\left(2 \, h - 1\right)} {\left(\frac{11 \, {\left(h - 7\right)}}{2 \, h - 1} - 6\right)}} \\ 0 & 0 & 1 & -\frac{14 \, a + 2 \, c - \frac{33 \, {\left(a h + b\right)}}{2 \, h - 1}}{3 \, {\left(\frac{11 \, {\left(h - 7\right)}}{2 \, h - 1} - 6\right)}} \end{array}\right)
(-a + 1/3*(14*a + 2*c - 33*(a*h + b)/(2*h - 1))*(2*(h - 7)/(2*h - 1) - 1)/(11*(h - 7)/(2*h - 1) - 6) + 2*(a*h + b)/(2*h - 1), 1/2*(a*h + b)/(2*h - 1) + 1/6*(14*a + 2*c - 33*(a*h + b)/(2*h - 1))*(h - 7)/((2*h - 1)*(11*(h - 7)/(2*h - 1) - 6)), -1/3*(14*a + 2*c - 33*(a*h + b)/(2*h - 1))/(11*(h - 7)/(2*h - 1) - 6)) 

a= var('k')
b= var('h')
c= var('c')
M = matrix(SR,[ [3,2], [h,k] ])
B= matrix(SR,[[-1],[7]])
M=M.augment(B,subdivide = True)
print "Starting M"
show(M)
print ""
print "Reduced M"
show(M.echelon_form())
Starting M
(32−1hk7)\displaystyle \left(\begin{array}{rr|r} 3 & 2 & -1 \\ h & k & 7 \end{array}\right)
Reduced M
(102 (h+21)3 (2 h−3 k)−1301−h+212 h−3 k)\displaystyle \left(\begin{array}{rr|r} 1 & 0 & \frac{2 \, {\left(h + 21\right)}}{3 \, {\left(2 \, h - 3 \, k\right)}} - \frac{1}{3} \\ 0 & 1 & -\frac{h + 21}{2 \, h - 3 \, k} \end{array}\right)

#PUT CURSOR ANYWHERE IN THIS BOX THEN HIT SHIFT+ENTER to start the applet
@interact
def _(row = slider([1..10], label='rows', default = 3) , col = slider([1..10], label='columns', default = 3)):

    @interact(layout=dict(top=[['M', 'b']]))

    def gauss_method(b= random_matrix(QQ,nrows = row, ncols = 1, algorithm='echelonizable', rank = 1), M=random_matrix(QQ,nrows = row, ncols = col, algorithm='echelonizable', rank = min(row,col)),rescale_leading_entry=True ):
        """Describe the reduction to echelon form of the given matrix of rationals.

        M  matrix of rationals   e.g., M = matrix(QQ, [[..], [..], ..])
        rescale_leading_entry=False  boolean  make the leading entries to 1's

        Returns: None.  Side effect: M is reduced.  Note: this is echelon form,
        not reduced echelon form; this routine does not end the same way as does
        M.echelon_form().

        """

        M=M.augment(b, subdivide = True)
        N=copy(M)
        N=N.rref()


        num_rows=M.nrows()
        num_cols=M.ncols()
        print("Reduced echelon form:")
        show(N)

        print("Steps")
        show(M)



        col = 0   # all cols before this are already done
        for row in range(0,num_rows):
            # ?Need to swap in a nonzero entry from below
            while (col < num_cols
                   and M[row][col] == 0):
                for i in M.nonzero_positions_in_column(col):
                    if i > row:
                        print " R",row+1," swap R",i+1
                        M.swap_rows(row,i)
                        show(M)
                        break
                else:
                    col += 1

            if col >= num_cols:
                break

            # Now guaranteed M[row][col] != 0
            if (rescale_leading_entry
               and M[row][col] != 1):
                print "",1/M[row][col],"R",row+1
                M.rescale_row(row,1/M[row][col])
                show(M)
            change_flag=False
            C=1
            for changed_row in range(row+1,num_rows):
                if M[changed_row][col] != 0:
                    change_flag=True
                    factor=-1*M[changed_row][col]/M[row][col]
                    print "",factor," R",row+1,"+R",changed_row+1
                    M.add_multiple_of_row(changed_row,row,factor)
                C +=1
            if change_flag:
                show(M)
            col +=1

        print "Above is the echelon form, let's keep cruising to get the reduced echelon form"

        for changed_row in range(1,num_rows):
            for row in range(0,changed_row):
                factor = -1*M[row][changed_row]
                if factor != 0:
                    print "",factor," R",row+1,"+R",changed_row+1
                    M.add_multiple_of_row(row,changed_row,factor)
                    show(M)
Interact: please open in CoCalc
︠2e1ea1f8-b719-43cd-8651-1e547ae09ce9︠




#PUT CURSOR ANYWHERE IN THIS BOX THEN HIT SHIFT+ENTER to start the applet
@interact
def _(row = slider([1..10], label='rows', default = 3) , col = slider([1..10], label='columns', default = 3)):

    @interact(layout=dict(top=[['M', 'b']]))

    def gauss_method(b= random_matrix(QQ,nrows = row, ncols = 1, algorithm='echelonizable', rank = 1), M=random_matrix(QQ,nrows = row, ncols = col, algorithm='echelonizable', rank = min(row,col)),rescale_leading_entry=True ):
        """Describe the reduction to echelon form of the given matrix of rationals.

        M  matrix of rationals   e.g., M = matrix(QQ, [[..], [..], ..])
        rescale_leading_entry=False  boolean  make the leading entries to 1's

        Returns: None.  Side effect: M is reduced.  Note: this is echelon form,
        not reduced echelon form; this routine does not end the same way as does
        M.echelon_form().

        """

        M=M.augment(b, subdivide = True)
        N=copy(M)
        N=N.rref()


        num_rows=M.nrows()
        num_cols=M.ncols()
        print("Reduced echelon form:")
        show(N)

        print("Steps")
        show(M)



        col = 0   # all cols before this are already done
        for row in range(0,num_rows):
            # ?Need to swap in a nonzero entry from below
            while (col < num_cols
                   and M[row][col] == 0):
                for i in M.nonzero_positions_in_column(col):
                    if i > row:
                        print " R",row+1," swap R",i+1
                        M.swap_rows(row,i)
                        show(M)
                        break
                else:
                    col += 1

            if col >= num_cols:
                break

            # Now guaranteed M[row][col] != 0
            if (rescale_leading_entry
               and M[row][col] != 1):
                print "",1/M[row][col],"R",row+1
                M.rescale_row(row,1/M[row][col])
                show(M)
            change_flag=False
            C=1
            for changed_row in range(row+1,num_rows):
                if M[changed_row][col] != 0:
                    change_flag=True
                    factor=-1*M[changed_row][col]/M[row][col]
                    print "",factor," R",row+1,"+R",changed_row+1
                    M.add_multiple_of_row(changed_row,row,factor)
                C +=1
            if change_flag:
                show(M)
            col +=1

        print "Above is the echelon form, let's keep cruising to get the reduced echelon form"

        for changed_row in range(1,num_rows):
            for row in range(0,changed_row):
                factor = -1*M[row][changed_row]
                if factor != 0:
                    print "",factor," R",row+1,"+R",changed_row+1
                    M.add_multiple_of_row(row,changed_row,factor)
                    show(M)
Interact: please open in CoCalc
#Example with variables in entries

a= var('a')
b= var('b')
c= var('c')
h= var('h')
M = matrix(SR,[ [-1,4,1], [h,-2,-7], [7,5,2] ])
B= matrix(SR,[[a],[b],[c]])
M=M.augment(B,subdivide = True)
print "Starting M"
show(M)
print ""
print "Reduced M"
show(M.echelon_form())
Starting M
(−141ah−2−7b752c)\displaystyle \left(\begin{array}{rrr|r} -1 & 4 & 1 & a \\ h & -2 & -7 & b \\ 7 & 5 & 2 & c \end{array}\right)
Reduced M
(100−a+(14 a+2 c−33 (ah+b)2 h−1)(2 (h−7)2 h−1−1)3 (11 (h−7)2 h−1−6)+2 (ah+b)2 h−1010ah+b2 (2 h−1)+(14 a+2 c−33 (ah+b)2 h−1)(h−7)6 (2 h−1)(11 (h−7)2 h−1−6)001−14 a+2 c−33 (ah+b)2 h−13 (11 (h−7)2 h−1−6))\displaystyle \left(\begin{array}{rrr|r} 1 & 0 & 0 & -a + \frac{{\left(14 \, a + 2 \, c - \frac{33 \, {\left(a h + b\right)}}{2 \, h - 1}\right)} {\left(\frac{2 \, {\left(h - 7\right)}}{2 \, h - 1} - 1\right)}}{3 \, {\left(\frac{11 \, {\left(h - 7\right)}}{2 \, h - 1} - 6\right)}} + \frac{2 \, {\left(a h + b\right)}}{2 \, h - 1} \\ 0 & 1 & 0 & \frac{a h + b}{2 \, {\left(2 \, h - 1\right)}} + \frac{{\left(14 \, a + 2 \, c - \frac{33 \, {\left(a h + b\right)}}{2 \, h - 1}\right)} {\left(h - 7\right)}}{6 \, {\left(2 \, h - 1\right)} {\left(\frac{11 \, {\left(h - 7\right)}}{2 \, h - 1} - 6\right)}} \\ 0 & 0 & 1 & -\frac{14 \, a + 2 \, c - \frac{33 \, {\left(a h + b\right)}}{2 \, h - 1}}{3 \, {\left(\frac{11 \, {\left(h - 7\right)}}{2 \, h - 1} - 6\right)}} \end{array}\right)
(-a + 1/3*(14*a + 2*c - 33*(a*h + b)/(2*h - 1))*(2*(h - 7)/(2*h - 1) - 1)/(11*(h - 7)/(2*h - 1) - 6) + 2*(a*h + b)/(2*h - 1), 1/2*(a*h + b)/(2*h - 1) + 1/6*(14*a + 2*c - 33*(a*h + b)/(2*h - 1))*(h - 7)/((2*h - 1)*(11*(h - 7)/(2*h - 1) - 6)), -1/3*(14*a + 2*c - 33*(a*h + b)/(2*h - 1))/(11*(h - 7)/(2*h - 1) - 6)) 

a= var('k')
b= var('h')
c= var('c')
M = matrix(SR,[ [3,2], [h,k] ])
B= matrix(SR,[[-1],[7]])
M=M.augment(B,subdivide = True)
print "Starting M"
show(M)
print ""
print "Reduced M"
show(M.echelon_form())
Starting M
(32−1hk7)\displaystyle \left(\begin{array}{rr|r} 3 & 2 & -1 \\ h & k & 7 \end{array}\right)
Reduced M
(102 (h+21)3 (2 h−3 k)−1301−h+212 h−3 k)\displaystyle \left(\begin{array}{rr|r} 1 & 0 & \frac{2 \, {\left(h + 21\right)}}{3 \, {\left(2 \, h - 3 \, k\right)}} - \frac{1}{3} \\ 0 & 1 & -\frac{h + 21}{2 \, h - 3 \, k} \end{array}\right)
#PUT CURSOR ANYWHERE IN THIS BOX THEN HIT SHIFT+ENTER to start the applet
@interact
def _(row = slider([1..10], label='rows', default = 3) , col = slider([1..10], label='columns', default = 3)):

    print "If you want a variables to appear use the syntax var('x'), var('y'), ... in the entries."
    @interact(layout=dict(top=[['M', 'b']]))

    def gauss_method(b= matrix(SR,[ [0] for i in range(row)]), M=matrix(SR,[ [0 for i in range(col)] for i in range(row)]),rescale_leading_entry=True ):
        """Describe the reduction to echelon form of the given matrix of rationals.

        M  matrix of rationals   e.g., M = matrix(QQ, [[..], [..], ..])
        rescale_leading_entry=False  boolean  make the leading entries to 1's

        Returns: None.  Side effect: M is reduced.  Note: this is echelon form,
        not reduced echelon form; this routine does not end the same way as does
        M.echelon_form().

        """

        M=M.augment(b, subdivide = True)
        N=copy(M)
        N=N.rref()


        num_rows=M.nrows()
        num_cols=M.ncols()
        print("Reduced echelon form:")
        show(N)

        print("Steps")
        show(M)



        col = 0   # all cols before this are already done
        for row in range(0,num_rows):
            # ?Need to swap in a nonzero entry from below
            while (col < num_cols
                   and M[row][col] == 0):
                for i in M.nonzero_positions_in_column(col):
                    if i > row:
                        print " R",row+1," swap R",i+1
                        M.swap_rows(row,i)
                        show(M)
                        break
                else:
                    col += 1

            if col >= num_cols:
                break

            # Now guaranteed M[row][col] != 0
            if (rescale_leading_entry
               and M[row][col] != 1):
                print "",1/M[row][col],"R",row+1
                M.rescale_row(row,1/M[row][col])
                show(M)
            change_flag=False
            C=1
            for changed_row in range(row+1,num_rows):
                if M[changed_row][col] != 0:
                    change_flag=True
                    factor=-1*M[changed_row][col]/M[row][col]
                    print "",factor," R",row+1,"+R",changed_row+1
                    M.add_multiple_of_row(changed_row,row,factor)
                C +=1
            if change_flag:
                show(M)
            col +=1

        print "Above is the echelon form, let's keep cruising to get the reduced echelon form"

        for changed_row in range(1,num_rows):
            for row in range(0,changed_row):
                factor = -1*M[row][changed_row]
                if factor != 0:
                    print "",factor," R",row+1,"+R",changed_row+1
                    M.add_multiple_of_row(row,changed_row,factor)
                    show(M)
Interact: please open in CoCalc

row = 4
col = 4

b= matrix(SR,[ [0] for i in range(row)])
M=matrix(SR,[ [0 for i in range(col)] for i in range(row)])
show(M,b)
(0000000000000000)\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) (0000)\displaystyle \left(\begin{array}{r} 0 \\ 0 \\ 0 \\ 0 \end{array}\right)