Exercise List all shapes of the row echelon forms of the augmented matrices of consistent systems of 2 equations in 2 variables with a unique solution. What about systems of other sizes?

Answer In this exercise, we have listed all the consistent cases for 2 equations and 2 variables. Among these, only the following one has a unique solution.

[ | * | # | # | ] |

0 | * | # |

For 3 equations in 2 variables, we are looking for 3 by 3 matrices with the first two columns pivot and the last column not pivot. The following is the only possibility.

[ | * | # | # | ] |

0 | * | # | ||

0 | 0 | 0 |

For 2 equations in 3 variables, we need a 2 by 4 matrix with first three columns pivot and the last column not pivot. Since there are only two rows, this is impossible. Consequently, no system of 2 equations in 3 variables can have a unique solution.

In general, for a system of equations to have unique solution, the number of equations cannot be less than the number of variables (see further discussion).