Gauss-Jordan elimination

I suppose you know what Gauss-Jordan elimination is, so the note here was just simply tell you why it can work.

As you know that, if I take my matrix A for example and I could chop it up, into four square blocks, suppose it's square, I just take a nice case, and B, suppose it's the same case as A, and then we can take the regular rule for this block multiplication.

Now let's jump back to the Gauss-Jordan elimination, I get the overall matrix E, that's the elimination matrix, the product of all those little steps the Gauss-Jordan elimination always do, then let E block times the augmented matrix, we get identity matrix I out of the left half, and the right half which was what we long to achieve, the inverse of matrix A. so that's the reason of Gauss-Jordan's idea that solve two equations at once can work.