# 213 - A7.5 - Finding exact inverses

## 231 days ago by Professor213

%hide %auto def RREF(A): reduced_echelon_form=True M = A.nrows() N = A.ncols() # if A.det()==0: # print html("<font size=+1 color=red>Matrix is Singular. Please pick another!</font>") 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,-2,-1],[-1,5,6],[5,-4,5]]) 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: All of the remaining rows are zero. Move to next columns or finish. After 3 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: All of the remaining rows are zero. Move to next columns or finish. After 3 step(s) the matrix is equivalent to: