What is the meaning of the messages that travel via procedures named by “np”? These travel via a sequence of such procedures towards older instances—from right to left. Each instance owns a value b which was its row just after dividing the row by its lead element—(car b) is one.
The message is (... x-1 one) and
conveys the coefficients of a linear combination
of columns to the right of the recipient’s own column that add to zero,
at least for those components beyond the one that concerns the recipient.
The coefficients are indeed
... x−2, x−1, 1
The rightmost, 1, is the coefficient for the column where singularity was discovered. The left most coeffieicnt is for the column immediately to the right of the column owned by the recipient.
The recipient is in a position to send on a new message, with one more leading scalar, describing an expanded linear combination of columns.
This will eventually be delivered to the caller of the inversion routine via the abnormal continuation. It can be taken as constructive proof that the matrix is singular.