####
The Additive Identity

The identity property of
addition states that when zero is added to any real number, the number
does not change. Thus, the number "0" is called the additive
identity for real numbers.

There is a matrix which is an *additive identity for matrices*:

This matrix is also denoted [0]. When [0] is added to any matrix of the
same dimensions, the matrix does not change. For example,

####
The Multiplicative Identity

The identity property of
multiplication states that when 1 is multiplied by any real number, the
number does not change; that is, any number times 1 is equal to itself.
The number "1" is called the multiplicative identity for real
numbers.

There is a matrix which is a multiplicative identity for matrices—the
identity matrix:

This matrix, denoted

*I*, is a square matrix. When any

*m*×*n*
matrix is multiplied
on the left by an

*m*×*m* identity matrix, or on the right by an

*n*×*n* identity
matrix, the

*m*×*n* matrix does not change. For example,

and