If A and
B are both m x n matrices,
then their sum is the m x n matrix
formed by adding the corresponding entries in each matrix.
If A = (aij)m x n and
B = (bij)m x n, then their sum is
A+B = (aij + bij)m x n
We can add only matrices with equal numbers of columns
and equal numbers of rows. For example, addition of a 3 x 2 matrix and
a 2 x 2 matrix is not defined.
Example. Add the matrices.
The m x n zero matrix, denoted by 0, is the m x n matrix with
each entry equal to zero. Since
A + 0 = 0 + A = A
for every m x n matrix A, the zero matrix is the additive identity for the
set of m x n matrices.
The additive inverse -A of the matrix A is (-1)A.
Thus,
A + (-A) = (-A) + A = 0
for any m x n matrix A.
Definition
.
To define subtraction of two m x n matrices A and B use the additive
inverse as folows:
A - B = A + (-B).
To subtract B from A we need only subtract the entries in B from the
corresponding entries in A.