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To perform the 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 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.
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:
To execute matrix multiplication...
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