Linear Algebra, Theory And Applications, 2012a

38 MATRICES AND LINEAR TRANSFORMATIONS
matrix which is obtained by adding corresponding entries. Thus
⎛
⎝ 1 2
⎞ ⎛
3 4 ⎠ + ⎝ −1 4
⎞ ⎛
2 8 ⎠ = ⎝ 0 6
5 12
5 2 6 −4 11 −2
Two matrices are equal exactly when they are the same size and the corresponding entries
are identical. Thus
⎛ ⎞
0 0 ( )
⎝ 0 0 ⎠ 0 0
≠
0 0
0 0
because they are different sizes. As noted above, you write (c ij ) for the matrix C whose
ij th entry is c ij . In doing arithmetic with matrices you must define what happens in terms
of the c ij 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 2.1.1 Let A =(a ij ) and B =(b ij ) be two m × n matrices. Then A + B = C
where
C =(c ij )
for c ij = a ij + b ij . Also if x is a scalar,
xA =(c ij )
where c ij = xa ij . The number A ij will typically refer to the ij th entry of the matrix A. The
zero matrix, denoted by 0 will be the matrix consisting of all zeros.
Do not be upset by the use of the subscripts, ij. The expression c ij = a ij + b ij is just
saying that you add corresponding entries to get the result of summing two matrices as
discussed above.
Note that there are 2 × 3 zero matrices, 3 × 4 zero matrices, etc. In fact for every size
there is a zero matrix.
With this definition, the following properties are all obvious but you should verify all of
these properties are valid for A, B, and C, m × n matrices and 0 an m × n zero matrix,
the commutative law of addition,
the associative law for addition,
the existence of an additive identity,
⎞
⎠ .
A + B = B + A, (2.1)
(A + B)+C = A +(B + C) , (2.2)
A +0=A, (2.3)
A +(−A) =0, (2.4)
the existence of an additive inverse. Also, for α, β scalars, the following also hold.
α (A + B) =αA + αB, (2.5)
(α + β) A = αA + βA, (2.6)
α (βA) =αβ (A) , (2.7)
1A = A. (2.8)
The above properties, (2.1) - (2.8) are known as the vector space axioms and the fact
that the m × n matrices satisfy these axioms is what is meant by saying this set of matrices
with addition and scalar multiplication as defined above forms a vector space.