Decide what operation was being performed in each case. To find the matrix that implements that operation, simply perform the same operation on the identity matrix. So, since step 1 swapped rows 2 and 3, your answer is correct. That E_1 matrix is I with rows 2 and three swapped.

Your E_2 is off though. The operation seems to be adding three times row 3 to row 1. If you do that to the identity matrix you get:

[1 0 3]

[0 1 0]

[0 0 1]

That’s the E_2 matrix. Your answer applied the row operation to E_1 instead of I, so your answer is actually the matrix product E_2 E_1. That’s the matrix that performs both of the first two row operations.

Multiplying row 2 by -1/2 in step 3 is an E_3 matrix of:

[ 1 . 0 . 0 ]

[ 0 -1/2 0 ]

[ 0 . 0 . 1 ]

Try that thinking to get E_4 on your own.

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