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## Main Question or Discussion Point

Hi, Everyone:

A question about finding the inverse of a matrix M using elementary

row operations (ERO's) E_k (where E_k is either a row-exchange, a scaling

of a row by k, or adding the multiple of one row to another row )

to do row-reduction in reduced-row-echelon

format, to end with the identity, which will have the form :

(E_k*E_(k-1)*....*E_2*E_1)*M=I

My doubt is that these ERO's may be done in different order, and

ERO matrices do not generally commute with each other. Also, the

process of row-reduction may be done in different ways: in some

cases, we may choose, e.g., to swap rows, and in other cases, we

may not. Bottom line is that we may arrive at the identity matrix

I in more than one way, i.e., by applying, in each case, different

sets of ERO's. As a specific example, I am thinking that, in one

choice, we may swap rows if the first row has leading coefficient

larger than 1 to avoid working with fractions, using matrices E_k. In another choice, we

may not swap rows, and work with fractions. In these two cases,

we are using different matrices E_k' to find the inverse of M. How

can we then guarantee that the products:

E_k*E_(k-1)*....*E_2*E_1

E_j' *E'_(j-1)*....*E_2'*E_1'

are equal to each other?

How can we then guarantee that the inverse matrix M that we

get this way-- As the product matrix E_k*E_(k-1)*....*E_2*E_1 is unique--as an inverse

must be?

Thanks in Advance.

A question about finding the inverse of a matrix M using elementary

row operations (ERO's) E_k (where E_k is either a row-exchange, a scaling

of a row by k, or adding the multiple of one row to another row )

to do row-reduction in reduced-row-echelon

format, to end with the identity, which will have the form :

(E_k*E_(k-1)*....*E_2*E_1)*M=I

My doubt is that these ERO's may be done in different order, and

ERO matrices do not generally commute with each other. Also, the

process of row-reduction may be done in different ways: in some

cases, we may choose, e.g., to swap rows, and in other cases, we

may not. Bottom line is that we may arrive at the identity matrix

I in more than one way, i.e., by applying, in each case, different

sets of ERO's. As a specific example, I am thinking that, in one

choice, we may swap rows if the first row has leading coefficient

larger than 1 to avoid working with fractions, using matrices E_k. In another choice, we

may not swap rows, and work with fractions. In these two cases,

we are using different matrices E_k' to find the inverse of M. How

can we then guarantee that the products:

E_k*E_(k-1)*....*E_2*E_1

E_j' *E'_(j-1)*....*E_2'*E_1'

are equal to each other?

How can we then guarantee that the inverse matrix M that we

get this way-- As the product matrix E_k*E_(k-1)*....*E_2*E_1 is unique--as an inverse

must be?

Thanks in Advance.