Proving BA=I using Elementary Row Operations and Determinants

[cocobaby]

Let A and B be 2x2 matrices s.t. AB=I . Then how can I prove that BA=I?


I assumed that there must exist some sequence of elementary row operations which carries B into I, and I denoted this sequence by the matrix A.

But here, I realized there's some pieces that I' m missing, which I colored red.

How can I explain it ? or is the way of proving this statement even valid?

Somebody help me please!

 

[radou]

If, for example, A is regular, then its inverse A^-1 = B and hence AB = BA = I. But, in general AB does not equal BA.

 

[Fredrik]

This looks like homework, so it should probably be in the homework forum.

I'll give you some hints:

1. Can you prove this if you know that at least one of the matrices is invertible?
2. Can you prove that at least one of them must be invertible?

 

[Redbelly98]

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[sponsoredwalk]

http://tutorial.math.lamar.edu/Classes/LinAlg/LinAlg.aspx

I believe the identity matrix section (i think) of this Linear Algebra text will help you out, I recall seeing a proof of this book somewhere.

 

[Fredrik]

It's really easy to do it the way I suggested, so I strongly suggesting that cocobaby try to do it that way instead of trying to find the proof in a book.

 

[HallsofIvy]

Together with Fredrick's suggestion, use the fact that det(AB)= det(A)det(B)= det(I)= 1.