Question

Can someone explain why in the equation E [AI]=IA^(-1), EA=I? I am referring to Prof. Strang's third lecture on MIT Open CourseWare. He mentions this at the end.

Answer 1

You are asking about Gauss-Jordan elimination on an augmented matrix?
E represents all the row operations that you perform on matrix A to convert it to the identity matrix I. In other words, by definition EA = I
In Lecture 2, he gave details on the elimination matrices, and how you can use E A to get I

Answer 2

Thank you! However, I am still a bit unclear about the notation. Does E [AI] mean that E is only applied to A (I guess, since you said E represents the operations performed to convert A to I it would make sense), but could you clarify me a bit on the notation?
I understand the Prof. Strang's explanation that precedes the notation, but when it comes to notation, I confuse matrix operation with number operation and the result is not the same.

Answer 3

The example lecture 3 is to find the inverse of matrix A
starting with the augmented matrix [ A I ]
let A=
1 3
2 7

augment with the identity matrix
1 3 | 1 0
2 7 | 0 1

now use elimination to zero out the 2 in location row 2, col 1
(using matrices, this would be equivalent to multiplying the A by
1 0
-2 1
and multiplying the I by this same "E" matrix

you get
1 3 | 1 0
0 1 | -2 1

(or you could treat the augmented matrix as a 2 x 4, multiply the first row by -2
and add that result to the 2nd row. You would get the same answer

now eliminate the 3 in (1,2)
which means multiply both sides by
1 -3
0 1

(equivalently, multiply the 2nd row by -3 and add to the first row in the 2 x 4 augmented matrix.

either way you get
1 0 | 7 -3
0 1 | -2 1

Answer 4

in that process, we call E = E2*E1
and we did
E A | E I
which is
E A | E

comparing that to our result
1 0 | 7 -3
0 1 | -2 1

we see EA matches up with the 2 x 2 identity matrix
and E matches the right-hand matrix
if you see EA= I , then you know E = inv(A)
(or A is the inverse of E)
thus we see that
7 -3
-2 1
(which is E) must be the inverse of A