6. Multiplication Of matrices : (Row by Column)
AB exists if, A = m x n & B = n x p
2 x 3 3 x 3
AB is matrix of 2×3
Note that, AB exists, bu BA does not Þ AB ¹ BA
(Number of columns in the pre multiplier = number of rows in post multiplier)
Note : In the product AB,
A= (a1,a2,..................an) & B =
1 × n n × 1
A B = [a1b1+a2b2+.........+anbn]
If A = [aij] be an m × n matrix & B = [bij] be an n × p matrix, then (AB)ij =r=1ån aij brj is a matrix of order m × p.
Proof:
Let A = [aij] be an m × n matrix and B [bij] be an n × p matrix. Then the m × p matrix C = [cij] is called the product if Cij = AiBj
where Ai is the ith row of A and Bj is the jth column of B. Thus the product AB is obtained as follows:
Thus (AB)ij = Ai Bj
= [ai1 b1j + ai2b2j + ...............+ ainbnj ]
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