Question

What is the best order to form the product ABC if A, B, and C are matrices with dimensions $3\times 9,9\times 4,4\times 2$, respectively.

Short answer

A(BC)

First multiply B and C, then multiply A with the obtained product BC.

Step-by-step solution

Step 1

Given:

A $3\times 9$matrix

B $9\times 4$matrix

C $4\times 2$matrix.

The matrix product ABC can be determined in two ways: $\left(AB\right)CorA\left(BC\right)$ (Note: The first way first multiplies matrices A and B, while the second way first multiplies B and C).

Multiplying a $p\times q$matrix and a $q\times r$matrix requires pqr multiplications.

AB requires $3\cdot 9\cdot 4=108$multiplications.

Product of AB and C requires $3\cdot 4\cdot 2=24$multiplications.

BC requires $9\cdot 4\cdot 2=72$multiplications.

Product of A and BC requires data-custom-editor="chemistry" $3\cdot 9\cdot 2=54$multiplications.

We then note that AB(C requires $108+24=132$, while AB(C) requires $72+54=126$multiplications and thus AB(C requires the least multiplications.