Is there another N×N matrix that works like I?
No. This is not quite obvious, but since 1 is unique for real numbers you might suspect that I is unique for NxN matrices.
Here is why this is so. In the following, assume that all the matrices have the correct dimensions for the multiplications. Suppose there is a matrix Z and that
ZA = A (1)
for any A with the right dimensions. The matrix Z works like I, but we are hoping that it will be different. We know how I works:
BI = B (2)
for any B with the right dimensions. Now replace A with I in (1) (since A can be any suitable matrix):
ZI = I (3)
Now replace B with Z in (2):
ZI = Z (4)
Looking at (3) and (4), Z and I are both equal to the same thing, so they must equal each other:
Z = I (5)
You can start at step (1) with the other order AZ = A and reach the same conclusion. So if you have found a matrix that works like the indentity, it is indeed the identity matrix.
What is the transpose of I?
1 0 0 0 1 0 0 0 1 T