For many reasons, doing the
matrix multiplication is better in this way:
The composition of the matrix
A sized
(m x
n) by the matrix
B sized
(n x
r) is the
matrix
C =
AB sized
(m x
r),
whose element
cij located at the
ijcell is equal to the sum of compositions of the elements of the
i-th line of the matrix
A by corresponding elements of the
j--th column of the matrix
B,i.e.
Multiplication of
A by
B is possible (composition of
AB can exist) if the number of the columns
A equals the number of the lines
B (in this case this two matrices are
consistent by form). Example:
There can be a composition
АВ sized
(m х
m), as well as composition
ВА sized
(m х
m),
for matrices
А(m х
n) and
В(n х
m).
It's clear that if
m is different from
n
these compositions can't be equal because of different resulting
matrices. But even if
m =
n,
i.e. in case of square matrices of equal order, compositions strong>
АВ and
ВА won't be necessary equal.
For example, for matrices
Therefore, the
matrix multiplication does not obey the commutative law
(АВ uneven
ВА)),
if
АВ =
ВА, then the matrices
A and
B are commute.
Associative and distributive laws of matrix multiplication aretrue in all cases where the dimensions of the matrix allows next steps:
(АВ)С = А(ВС) = ABC (associativity),
А(В + С) = АВ + АС (distributivity of multiplication to the left of addition process)
(А + В)С = АС + ВС (distributivity of multiplication to the right of addition process).
Multiplication of the
(m x
n)-matrix
А by the identity matrix of
m-th
order to the left and by the identity matrix of
n-th order to the right does not change the matrix, i.e.
ЕmА = АЕn = A.
If at least one of the matrices of the composition
АВ is zero, the result will be a zero matrix.
Note that if
АВ = 0 it doesn't mean that
А = 0 или
В = 0.
This can be seen in the following example:
