Is there another N×N matrix that works like I?

A good answer might be:

No. This is not quite obvious, but since 1 is unique you might suspect that I is unique.

I is Unique

It is easy to show that this is so. Suppose you had a matrix Z and that

(1) ZA = A

for any A. We know how I works:

(2) BI = B

for any B. Now use (1) with I replacing A:

(3) ZI = I

Now use (2), substituting Z for B:

(4) ZI = Z

Looking at (3) and (4):

(4) Z = I

It is also true that it works with the other order: AZ = A.


What is the transpose of I?

1  0  0
0  1  0
0  0  1