math111_logo Existence


Exercise List all shapes of the row echelon forms of 2 by 2 matrices A so that Ax = b has solutions for any b. What about matrices of other sizes?

Answer We look for row echelon forms with no nonzero row (i.e., all rows pivot). For the 2 by 2 coefficient matrices, there is only one possibility.

[ * # ]
0 *

For the 2 by 3 coefficient matrices, we have the following possibilities.

[ * # # ]
0 * #
[ * # # ]
0 0 *
[ 0 * # ]
0 0 *

We cannot find 3 by 2 matrices with all rows pivot. (This means that any system of 3 equations in 2 variables is inconsistent for some right side.)

In general, for a system of equations to be always consistent, the number of equations cannot be more than the number of variables (see further discussion).