determinant and invertibility for matrices I'm reading this in my text:

I don't understand this at all: (see attachment)
It's the second part that I don't get. I get that if B has a row of 0s, then the determinant is 0 (I could cofactor expand this and it'd be 0). But here's what I don't get:


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*if $|B| = 0$, why does this imply that $|A| = 0$? Where does this come from?

*if A is row equivalent to I, why does this mean that A is invertible?

Note that every elementary row operation is a multiplication of your matrix by an invertible matrix.
So if you can reduce your matrix to the identity matrix via elementary row operations that means you have multiplied your matrix by some invertible matrices to get $I$, therefore the  product of those matrices is the inverse of your matrix.
Thus your matrix in invertible. 
