213 - A7.5 - Finding exact inverses

55 days ago by Professor213

%hide %auto def RREF(A): reduced_echelon_form=True M = A.nrows() N = A.ncols() # if A.det()==0: # pretty_print(html("<font size=+1 color=red>Matrix is Singular. Please pick another!</font>")) pretty_print(html("<font size=+2 color=blue>Using the matrix</font>")) show(A) global Astart Astart = copy(A) # Now, let's start the step by step Gauss-Jordan routine ROWS = range(M) COLS = range(N) PIVOTS = range(min(M,N)) for i in PIVOTS: # Gauss-Jordan on column i # First, make the pivot element equal to 1 k = i # Need to fix this when a column is skipped when finding nonzero pivots # That is, if the remainder of a column is zero, go on to the next column till you reach the end. if A[i][i]==0: # swap for non-zero pivot # start looking in rows below the i-th while (k<M and A[k][i]==0): k += 1 if k==M: print 'All of the remaining rows are zero. Move to next columns or finish.' else: A.swap_rows(i,k) # So, now we should have a non-zero in the pivot position or k=M which means we are done. # Should consider using the function "nonzero_positions_in_column(i)" and use the min position > i. # Start zeroing out the rest of column i. if A[i][i]<>0: A[i]=A[i]/A[i][i] if reduced_echelon_form: for k in ROWS: if i<>k: A[k] = A[k]-A[i]*A[k][i] else: for k in ROWS: if i<k: A[k] = A[k]-A[i]*A[k][i] print '\n After',i+1,'step(s) the matrix is equivalent to:' show(A) 
       
num = 10 A = random_matrix(QQ, num, num) show(A) Ainv = A.inverse() show(Ainv) show(Ainv*A) show(A*Ainv) 
       

                                
                            

                                
# Here you can enter your own matrix or use the random one from above. # Comment this out if using the one from above A = matrix(QQ,[[1,0,-1,0],[0,-1,5,5],[6,-4,4,0],[0,0,1,1]]) show(A) 
       

                                
                            

                                
Aug = A.augment(identity_matrix(A.nrows())) show(Aug) 
       

                                
                            

                                
RREF(Aug) 
       
Using the matrix

 After 1 step(s) the matrix is equivalent to:

 After 2 step(s) the matrix is equivalent to:

 After 3 step(s) the matrix is equivalent to:

 After 4 step(s) the matrix is equivalent to:
Using the matrix

 After 1 step(s) the matrix is equivalent to:

 After 2 step(s) the matrix is equivalent to:

 After 3 step(s) the matrix is equivalent to:

 After 4 step(s) the matrix is equivalent to:
A = matrix(QQ,[[1, 0, -1, 0 ],[ 0, -1, 5 ,5],[ 6, -4, 4 ,0],[ 0, 0, 1, 1]]) Aug = A.augment(identity_matrix(A.nrows())) show(A) 
       

                                
                            

                                
RREF(Aug) 
       
Using the matrix

 After 1 step(s) the matrix is equivalent to:

 After 2 step(s) the matrix is equivalent to:

 After 3 step(s) the matrix is equivalent to:

 After 4 step(s) the matrix is equivalent to:
Using the matrix

 After 1 step(s) the matrix is equivalent to:

 After 2 step(s) the matrix is equivalent to:

 After 3 step(s) the matrix is equivalent to:

 After 4 step(s) the matrix is equivalent to:
B = block_matrix(QQ,[[A,identity_matrix(4)]]) RREF(B) 
       
Using the matrix

 After 1 step(s) the matrix is equivalent to:

 After 2 step(s) the matrix is equivalent to:

 After 3 step(s) the matrix is equivalent to:

 After 4 step(s) the matrix is equivalent to:
Using the matrix

 After 1 step(s) the matrix is equivalent to:

 After 2 step(s) the matrix is equivalent to:

 After 3 step(s) the matrix is equivalent to:

 After 4 step(s) the matrix is equivalent to:
A = matrix(QQ,[[1, 1, 1, 0 ],[ 0, -2, 8, 0],[ 5, 3, 11 ,1],[ 0, 0, 2, -2]])