**IA** = **A**

The matrix **I** is called an **identity matrix** because
**IA** = **A** and **AI** = **A** for all matrices **A**.
This is similar to the number **1**,
which is called the multiplicative identity,
because 1a = a and a1 = a for all real numbers a.

There is no matrix that works as an identity for
matrices of all dimensions.
For **N×N** square matrices
there is a matrix **I**_{N×N}
that works as an identity.

If **I**_{N×N} is the N-dimensional identity matrix,
then its elements
are 1 on the main diagonal and 0 elsewhere.

1 0 0 0 1 0 0 0 1The

main diagonalconsists of those elements [I]_{m m}where the row number is equal to the column number. At left is an example of an identity matrix.

The 1s must run down that particular diagonal; it won't work with the other diagonal.