Use matrix multiplication in an application
In an application product, the shared category is the columns of the first matrix and rows of the second. Each product entry is a row-by-column total, and its labels come from the two outer categories
\[\left(AB\right)_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}\]
- Put the shared category inside The shared category must be the columns of the first matrix and the rows of the second matrix \(\displaystyle A_{r\times n}B_{n\times c}\Rightarrow AB\text{ has order }r\times c\)
- Use a row and a column Each product entry pairs one row of the first matrix with one column of the second matrix \(\displaystyle (AB)_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+a_{i3}b_{3j}\)
- Add every paired product The products for all shared categories are added to form one entry \(\displaystyle 3\times4+2\times6+4\times3=36\)
- Read the outer labels The product rows keep the row labels of the first matrix and the product columns keep the column labels of the second matrix \(\displaystyle AB:\quad\text{rows from }A,\quad\text{columns from }B\)
Worked example
A training centre records the number of sessions completed by two groups in three activities. Matrix A gives the counts, and column matrix B gives the points for one session in each activity. Find the total points for each group
\[A=\begin{bmatrix}2&4&1\\5&1&3\end{bmatrix},\quad B=\begin{bmatrix}6\\3\\8\end{bmatrix}\]
- \(\displaystyle A:2\times3,\quad B:3\times1,\quad AB:2\times1\)The three activities are the shared dimension, so the product is defined and has two rows and one column
- \(\displaystyle (AB)_{11}=2\times6+4\times3+1\times8=12+12+8=32\)The first row gives the three contributions for the first group, which total 32 points
- \(\displaystyle (AB)_{21}=5\times6+1\times3+3\times8=30+3+24=57\)The second row gives the three contributions for the second group, which total 57 points
- \(\displaystyle AB=\begin{bmatrix}32\\57\end{bmatrix}\)The product has one entry for each group, so the totals are 32 and 57 points
Common mistakes
Using the factors in the wrong order
✕\(\displaystyle B_{3\times1}A_{2\times3}\text{ gives the required totals}\)✓\(\displaystyle A_{2\times3}B_{3\times1}\text{ gives the required totals}\)
The shared category must be the first matrix's columns and the second matrix's rows so that the row-column products form the required totalUsing only one paired product
✕\(\displaystyle (AB)_{11}=2\times6=12\)✓\(\displaystyle (AB)_{11}=2\times6+4\times3+1\times8=32\)
Every shared category contributes a paired product, and all of those products must be addedMislabeling a product entry
✕\(\displaystyle (AB)_{11}=\text{activity 1 total for Group 1}\)✓\(\displaystyle (AB)_{11}=\text{total points for Group 1}\)
The product's row label comes from the first matrix and its column label comes from the second matrix, while the shared activity category is summed outDifficulty levels
introMultiply a two-row quantity matrix by a single column of values
standardUse a two-column second factor to calculate more than one labelled quantity
challengeBuild both factors from a description and interpret an entry using the two outer labels
examChoose the product order or the entry that gives the stated application total
Questions about this topic
Why does the quantity matrix go first in this application?+
Its rows identify the quantities being totalled and its columns match the rows of the per-unit value matrix, so the shared category is summed over
What does one entry of an application product represent?+
It is the sum of the paired products for one outer row category and one outer column category
Why is matrix multiplication not treated as entry-by-entry multiplication?+
Each product entry uses a complete row and column, multiplying matching shared categories and adding their contributions
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 4: Matrices
In this chapter
State the order of a matrix and locate its elements4A Matrix types and order: Identify row, column, square, symmetric and zero matricesRead information from a labelled data matrix4B Matrix addition and subtraction: Identify row, column, square, symmetric and zero matricesAdd and subtract matrices4B Matrix addition and subtraction: Find unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matricesDecide whether a matrix product is defined and give its orderMultiply matrices4D Matrix products and powers: Evaluate powers of a square matrixIdentify and construct the identity matrix4E Inverse matrices and simultaneous equations: Find unknown entries in a matrix equationFind the determinant of a 2 × 2 matrixFind the inverse of a 2 × 2 matrixWrite simultaneous equations as a matrix equationSolve simultaneous equations using the inverse matrix4F Matrices for networks: Evaluate powers of a square matrixRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixSet up a transition matrixUse a transition matrix to predict the next step4H State matrices and recurrence: Evaluate powers of a square matrixGenerate state matrices by recursionDetermine the long-run state of a transition modelUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totals