130 CHAPTER 8. MATRICES
consider the following example in which a matrix is being multiplied by the scalar 3.
3
1 2 3 45 2 8 76 −9 1 2
=
3 6 9 1215 6 24 2118 −27 3 6
.
The new matrix is obtained by multiplying every entry of the original matrix by the givenscalar. If A is an m×n matrix, −A is defined to equal (−1)A.
Two matrices must be the same size to be added. The sum of two matrices is a matrixwhich is obtained by adding the corresponding entries. Thus 1 2
3 45 2
+
−1 42 86 −4
=
0 65 1211 −2
.
Two matrices are equal exactly when they are the same size and the corresponding entriesare identical. Thus 0 0
0 00 0
̸=( 0 00 0
)
because they are different sizes. As noted above, you write (ci j) for the matrix C whosei jth entry is ci j. In doing arithmetic with matrices you must define what happens in termsof the ci j sometimes called the entries of the matrix or the components of the matrix.
The above discussion stated for general matrices is given in the following definition.
Definition 8.1.1 (Scalar Multiplication) If A = (ai j) and k is a scalar, then kA = (kai j) .
Example 8.1.2 7
(2 01 −4
)=
(14 07 −28
).
Definition 8.1.3 (Addition) If A=(ai j) and B=(bi j) are two m×n matrices. Then A+B=C where
C = (ci j)
for ci j = ai j +bi j.
Example 8.1.4 (1 2 31 0 4
)+
(5 2 3−6 2 1
)=
(6 4 6−5 2 5
)
To save on notation, we will often use Ai j to refer to the i jth entry of the matrix A.
Definition 8.1.5 (The zero matrix) The m×n zero matrix is the m×n matrix having everyentry equal to zero. It is denoted by 0.
Example 8.1.6 The 2×3 zero matrix is
(0 0 00 0 0
).