**ADDITIVE INVERSE**

** **

The **additive inverse **(or **negative**) of a matrix – *X *is the matrix in which

each element is the additive inverse of the corresponding element of *X*.

If

and B =

then by the definition of the additive inverse of a matrix,

and -B =

By the definition of matrix addition, for each matrix *X *the sum X + (-X) is

a **zero matrix, ***O*, whose elements are all zeros. For the matrix *A *above,

A – A = .

There is an m n zero matrix for each pair of values of m and n. Such a matrix serves as an m n additive identity, similar to the additive identity 0 for any real number. Zero matrices have the following identity property.

**Zero Matrix**

If O is an m n zero matrix, and A is any m n matrix, then

A + O = O + A = A.