# Row-reduced echelon matrices

1. Consider the following systems of equation
Find out whether they have solutions. If so, describe explicitly all solutions.
2. Let

for which triples does the system have a solution?
3. Let

for which does the system have a solution?
4. Suppose and are row-reduced echelon matrices and that the systems and have exactly the same solutions. Prove that
5. Let

Find a row-reduced echelon matrix which is row-equivalent to and an invertible matrix such that .
6. For each of the three matrices

use elementary row operations to discover whether it is invertible, and to find the inverse in case it is.
7. Suppose is a matrix and that is a matrix. Prove that is not invertible.
8. Let be an matrix. Prove the following:
1. If is invertible and for some matrix , the .
2. If is not invertible, then there exists an matrix such that but .
9. Let

Prove, using elementary row operations, that is invertible if and only if .
10. Let

For which does there exist a scalar such that ?
11. An matrix is upper-triangular if for . Prove that is invertible if and only if every entry on its main diagonal is distinct from 0.
12. Prove that if is an matrix, is an matrix and , then is not invertible (generalization of a previous problem).
13. Let be an matrix. Show that by means of a finite number of elementary row and/or column operations one can pass from to a matrix which is both row-reduced echelon and column-reduced echelon, i.e., if , , . Show that where is an invertible matrix and is an invertible matrix.
Subhashis Banerjee 2009-09-03