Identity Matrix: Is Inverse Always True for n>=2?

[mathmathmad]

Homework Statement

let I_n be as an identity matrix where a_ij = 1 when i=j
I just want to ask that is it true that all identity matrix has an inverse (determinant is not 0) for n>=2?

The Attempt at a Solution

 

[CompuChip]

The determinant of I_n is 1, as you can easily compute (expanding along any row or column, you find that det(I_n) = 1 . det(I_{n - 1}) and clearly det(I₁) = 1).

Also it is invertible, and it is its own inverse. You can check this directly from the definition.