Decide whether a matrix product is defined and give its order

Order of Q

\(\displaystyle 3\times 4\)

Order of P

\(\displaystyle 4\times 2\)

A bakery keeps an orders matrix Q for 3 shops and 4 products, and a price matrix P for 4 products and 2 price lists. What is the order of the product QP?

One entry of AB pairs a row of A with a column of B entry by entry. Make the two tile rows match, then make them differ, and watch the last two readouts

When did AB exist, and what order did it have for every setting of the sliders?

Defined products and product order

A matrix product is defined only when the inner dimensions match. Once it is defined, the outer dimensions give the order of the product

1. 1See the matching row and column lengths Here each row of A has three entries and each column of B has three entries, so they can be paired. Two rows of A and two columns of B produce a two-by-two product \(\displaystyle A=\begin{bmatrix}1&2&0\\3&1&4\end{bmatrix},\qquad B=\begin{bmatrix}2&1\\0&3\\1&2\end{bmatrix}\)
2. 2Compare the inner numbers For QP, compare the 4 columns of Q with the 4 rows of P \(\displaystyle Q_{3\times 4}P_{4\times 2}:\quad 4=4\)
3. 3Use the written order QP and PQ use different inner pairs, so check the factors as written \(\displaystyle QP\text{ and }PQ\text{ must be checked separately}\)
4. 4Take the outer numbers When the product is defined, its rows come from the first matrix and its columns come from the second matrix \(\displaystyle Q_{3\times 4}P_{4\times 2}\text{ has order }3\times 2\)
5. 5Check before calculating An unequal inner pair means no product is formed, whatever the entries are \(\displaystyle P_{4\times 2}Q_{3\times 4}:\quad 2\ne 3\Rightarrow PQ\text{ is undefined}\)

Order of H

\(\displaystyle 5\times 3\)

Order of R

\(\displaystyle 3\times 2\)

A sports club keeps an hours matrix H for 5 members and 3 activities, a rates matrix R for 3 activities and 2 seasons, How many entries does the product HR contain?

Check the inner numbers of H and R first, then take the outer numbers and count the entries in that order

Order of H

\(\displaystyle 5\times 3\)

Order of R

\(\displaystyle 3\times 2\)

Order of F

\(\displaystyle 1\times 5\)

How many entries does the product FHR contain?

Check F with H first, then the result with R. Take the outer numbers from the ends of the chain

Order of S

\(\displaystyle 2\times 5\)

Order of T

\(\displaystyle 2\times 3\)

Which line correctly tests whether ST is defined?