# Elementary matrix

Please kindly show each step of your solution. Thank you.

Without using Proposition 2.9, show that for any elementary matrix E...

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Please see the attachment.

Proof:

We consider three types of elementary matrix.

We start from the identity matrix .

Type 1: is to interchange two rows of . Suppose it interchanges row i and row j, then .

Type 2: is to add a multiple of one row to another. In this case, is an upper(or lower) triangle matrix with all diagonal entry 1. Then is an lower(or upper) triangle matrix with all diagonal entry 1. Thus .

Type 3: is to multiply any row by a nonzero element . In this case, is an diagonal matrix with all diagonal entry 1 except one entry . So and we have .

Now for a general elementary matrix , it can be expressed as the multiplication of the above three types of matrix. Assume , then we have

Therefore, for any elementary matrix, we always have .

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