Let A = [ahi], B = [bij], and C = [cjk] be matrices of sizes m × n, n × p, and p × q, respectively. To establish the associative law A(BC) = (AB) C, proceed as follows. By Equation (16) the hj th element of AB is
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By another application of Equation (16), the hkth element of (AB) C is
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Show similarly that the double sum on the right is also equal to the hkth element of A(BC). Hence the m × q matrices (AB) C and A(BC) are equal.
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