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.