How to Use Matrices to Represent Data

Build a matrix from a data table

Choose what each row and column represents, keep that order fixed, and put each number in its matching position. A matrix is the rectangular array of numbers; the row labels, column labels and units tell you what those numbers mean.

Suppose three school clubs record the number of notebooks and pencils they collect. These are invented values for a worked example.

Use rows in the order Art, Science, Drama and columns in the order Notebooks, Pencils. The data matrix is

\(A=\begin{bmatrix}12 & 18\\9 & 15\\6 & 10\end{bmatrix}.\)

There are three rows and two columns, so A is a 3 × 2 matrix. The club names and item names remain in your key; they are not extra numerical entries.

Read an entry in context

In \(a_{ij}\), the first subscript gives the row and the second gives the column. Thus \(a_{21}=9\) means Science collected 9 notebooks. By contrast, \(a_{12}=18\) means Art collected 18 pencils. Swapping the subscripts changes which value you read.

The second row, [9, 15], describes Science’s collection. The second column, [18, 15, 10], describes pencils across all three clubs. The pencil total is 18 + 15 + 10 = 43. Read the labels before deciding whether a row total or a column total answers your question.

You may instead put clubs in columns and item types in rows. That gives \(A^T=\begin{bmatrix}12 & 9 & 6\\18 & 15 & 10\end{bmatrix}\), a 2 × 3 matrix containing the same information with a different orientation. Update the labels whenever you transpose the matrix.

Check the labels before calculating

• Keep one ordering. If another day’s matrix lists Science first, reorder its rows before combining matching club totals.
• Keep units clear. A column measured in centimeters is different from one measured in kilograms. A rectangular matrix can store both, but adding those two measurements would not produce a meaningful total.
• Distinguish zero from missing data. Zero notebooks means none were collected; an unrecorded count is unknown. Do not silently replace an unknown value with zero.
• Check the shape. Every row must contain one entry for every column category.

When two matrices use the same clubs, item categories and units, you can add their corresponding entries to combine the counts. For more connected lessons, use the matrix topics in the Algebra 2 Online Center.

Curriculum connection: Common Core HSN-VM.C.6 includes representing and manipulating data with matrices. The example and explanations above are original to this lesson.