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Find the possible failures in the column and the row picture

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.

Solution Preview

Please see the attached file.

Solution. Let's consider all of the possible cases we can have in a 3 × 3 system.
1) Success. In order for a system to be successful through elimination, there may be two possibilities for the structure of the coefficient matrix.

a) Full Success. The matrix may contain three independent columns. This way, the columns span all of space, and the planes formed by the rows intersect at a single point. Example:

b) Temporary Failure. The matrix may need a row exchange to allow for a pivot to be found. After this, three ...

Solution Summary

All possible structures of a 3x3 system are examined. Two possibilities for success are defined, as well as two possibilities for failure.