Find the possible failures in the column picture and the row picture, and match them up. Success would be 3 columns whose combinations give every vector b, which matches with 3 planes in the row picture that intersect at one point (the unique solution x). Give numerical examples of these two types of failure:
a. 3 columns lie on the same line; 3 planes are parallel (then if b happens to lie on that line of columns, the 3 planes meet in a ... )
b. 3 columns in the same plane, but no two on the same line. Then 3 planes do what? Which b's are okay?
Now give numerical examples of the other types of failure in the column and row pictures.
All possible structures of a 3x3 system are examined. Two possibilities for success are defined, as well as two possibilities for failure.