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