"From lecture 1 page 12, should E3 be 1 0 -3/2 1/2 not 1 0 0 1/2?"

No it is correct. I think my layout confused you because normally we put the row step before the matrix.

In this example I put the row step we used after the matrix so it could be next to the corresponding elementary matrix.

Have a look at the picture in the file below and hopefully it will be clear.