Let be a field and let be the set of all matrices with entries from Let be the identity matrix and let be the matrix whose -entry is and all other entries zero. We will denote by the diagonal matrix whose -entry is

**Definition**. For every and define We will call an **elementary matrix.**

**Remark**. If then clearly multiplying on the left by takes and adds on times the -th row of to the -th row of Similarly multiplying on the right by takes and adds on times the -th column of to the -th column of

**Problem 1**. Prove that

1)

2)

**Solution**. The first part is obvious because is a triangular matrix and so its determinant is the product of its diagonal entries, which are all To prove the second part of the problem, note that since we have and thus

**Problem 2**. Let be a diagonal matrix with Prove that is a product of of elementary matrices.

**Solution**. Let Since we have

In partcular, all are non-zero and hence invertible in See that

As you see we have multiplied on the left by elementary matrices and we got a diagonal matrix whose last diagonal entry is If we continue by induction, we will get a matrix which is the product of elementary matrices and

by Thus and the result follows from the second part of Problem 1.