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Operations on Matrices - Multiplication of Matrices

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Non-commutativity of multiplication of matrices, Zero matrix as the product of two non zero matrices


The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. Let A = `[a_(ij)]` be an m × n matrix and B = `[b_(jk)]` be an n × p matrix. Then the product of the matrices A and B is the matrix C of order m × p. To get the `(i, k)th`element cik of the matrix C, we take the ith row of A and kth column of B, multiply them elementwise and take the sum of all these products. In other words, if A = `[a_(ij)]_(m × n)`, B = `[b_(jk)]_(n × p)`, then the ith row of A is `[a_(i1)  a_(i2)  ...  a_("in")]` and the `k^(th)` column of

B is `[(b_(1k)) , (b_(2k)), (b_(nk))]` 
, then `c_(ik) =  a_(i1) b_(1k) + a_(i2) b_(2k) + a_(i3) b_(3k) + ... + a_("in") b_(nk)`   
=\[\displaystyle\sum_{j=1}^{n} a_{ij} b_{jk}\].

Non-commutativity of multiplication of matrices:
The below example that even if AB and BA are both defined, it is not necessary that AB = BA.
If A = `[(1,0),(0,-1)]` and B =`[(0,1),(1,0)]` ,

then AB `[(0,1),(-1,0)]`

and BA = `[(0,-1),(1,0)]`.
Clearly AB ≠ BA.
Thus matrix multiplication is not commutative.

Zero matrix as the product of two non zero matrices:
The real numbers a, b if ab = 0, then either a = 0 or b = 0. This need not be true for matrices, we will observe this through an example.
find AB ,if A = `[(0,-1),(0,2)]` and B = `[(3,5),(0,0)]`

We have AB = `[(0,-1),(0,2)][(3,5),(0,0)]`

Thus, if the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.

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