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5.2. MATRICES AND DETERMINANTS 169

The difference of two matrices A ≡ [a ij ]and B ≡ [b ij ]ofthe same size m × n is the
matrix C ≡ [c ij ]ofsize m × n with entries

c ij = a ij – b ij (i = 1, 2, ... , m; j = 1, 2, ... , n).
The difference of two matrices is denoted by C = A – B, and the operation is called
subtraction of matrices.

The product of a matrix A ≡ [a ij ]of size m × n by a scalar λ is the matrix C ≡ [c ij ]of
size m × n with entries

c ij = λa ij (i = 1, 2, ... , m; j = 1, 2, ... , n).
The product of a matrix by a scalar is denoted by C = λA, and the operation is called
multiplication of a matrix by a scalar.
Properties of multiplication of a matrix by a scalar:
0A = O (property of zero),
(λμ)A = λ(μA) (associativity with respect to a scalar factor),
λ(A + B)= λA + λB (distributivity with respect to addition of matrices),
(λ + μ)A = λA + μA (distributivity with respect to addition of scalars),
where λ and μ are scalars, matrices A, B, C, and zero matrix O have the same size.
The additively inverse (opposite) matrix for a matrix A ≡ [a ij ]of size m×n is the matrix
C ≡ [c ij ]ofsize m × n with entries

c ij =–a ij (i = 1, 2, ... , m; j = 1, 2, ... , n),
or,inmatrixform,
C =(–1)A.
Remark. The difference C of two matrices A and B can be expressed as C = A +(–1)B.
The product of a matrix A ≡ [a ij ]ofsize m × p and a matrix B ≡ [b ij ]ofsize p × n is
the matrix C ≡ [c ij ]ofsize m × n with entries
p

c ij = a ik b kj (i = 1, 2, ... , m; j = 1, 2, ... , n);
k=1
i.e., the entry c ij in the ith row and jth column of the matrix C is equal to the sum of
products of the respective entries in the ith row of A and the jth column of B. Note that
the product is defined for matrices of compatible size; i.e., the number of the columns in
the first matrix should be equal to the number of rows in the second matrix. The product of
two matrices A and B is denoted by C = AB, and the operation is called multiplication of
matrices.
Example 1. Consider two matrices
1 2 0 10 1
A = and B = .
6 –3 –6 –0.5 20
The product of the matrix A and the matrix B is the matrix
1 2 0 10 1

C = AB =
6 –3 –6 –0.520
1×0 + 2× (–6) 1×10 + 2× (–0.5)
1× 1 + 2× 20 –12 9 41
= = .
6× 0 +(–3) × (–6) 6× 10 +(–3) × (–0.5) 6×1 +(–3) ×20 18 61.5 –54
Twosquare matricesA and B are saidtocommute if AB =BA, i.e., if their multiplication
is subject to the commutative law.