Just a quick question on the identity matrix

[Jyupiter]

I'm just wondering, is an identity matrix, say I₃ considered as an elementary matrix? It's obviously possible, since we can multiply any row of I with a constant 1. I'm just curious if there is a restriction for rescaling with a constant 1.

 

[HallsofIvy]

Yes, the identity matrix is an elementary matrix (perhaps the most elementary!). An "elementary matrix" is any matrix that is derived from the identity matrix by doing anyone of the three "row-reduction" steps:
1) Multiply every number in a single row by a constant.
2) Swap two rows.
3) Replace one row by itself plus or minus a constant times another row.

You can get the identity matrix from the matrix by, as you say "multiply every number in one row by the constant 1" or by "replace one row by itself plus or minus 0 times another row".

 

[Jyupiter]

Thanks for the quick reply! What you said made sense. About the self-swapping row operation though, Hefferon's text stated that it's not allowed (said it has something to do with determinants later; I'm not that far into LA yet!). Is that restriction not the convention?

 

[HallsofIvy]

I've never heard of such a restriction. If you are talking about determinants (elementary matrices and row-reduction are used for much more) then swapping two rows just multiplies the determinant by -1.

 

[Jyupiter]

I'm sorry for being unclear in my previous reply. What I meant by "self-swapping" is replacing a row in a matrix by itself, e.g. Row 2 swaps with Row 2. I have no idea what determinants are yet, but perhaps its because of this that "self-swapping" is not allowed? As you could always arbitrarily replace a row by itself and as a result you'll get a different answer... In any case, I apologize if I am being a tad bit confusing!