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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 A sized (m x n) by the matrix
B sized (n x r) is the matrix
C = AB sized (m x r),
whose element c_{ij} located at the ijcell is equal to the sum of compositions of the elements of the ith line of the matrix
A by corresponding elements of the jth column of the matrix B,i.e.
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
we've got:
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 mth order to the left and by the identity matrix of nth order to the right does not change the matrix, i.e. Е_{m}А = АЕ_{n} = 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:
To execute matrix multiplication... To the list of possible tasks...  

