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8.2 I Operations with Matrices
609
One important property of addition of real numbers is that the number 0 is the
additive identity. That is, c + 0 = c for any real number c. For matrices,
a similar property holds. That is, if A is an m x n matrix and O is the m x n
zero matrix consisting entirely of zeros, then A + O — A.
In other words, O is the additive identity for the set of all m x n matrices. For
example, the following matrices are the additive identities for the set of all
2 x 3 a n d 2 x 2 matrices.
0
0
O =
Note The algebra of real numbers
and the algebra of matrices also have
important differences, which will be
discussed later.
and
O =
Zero 2 x 3 matrix
0
Zero 2 x 2 matrix
The algebra of real numbers and the algebra of matrices have many similarities.
For example, compare the following solutions.
Real Numbers
(Solve for x.)
m x n Matrices
(Solve for X.)
x +a = b
X +A =B
x + a + (-a) = b + (-a)
x +0 = b - a
x =b - a
X + A + (-A) = B + (-A)
X +O= B - A
X = B- A
EXAMPLE 5
^
Solving a Matrix Equation
Solve for X in the equation 3X + A ~ B, where
o
Solution
Begin by solving the equation for X to obtain
3X = B - A
X =
(B - A).
Now, using the matrices A and B, you have
1
2
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