Chapter 8: Apply matrices

8.1 Introduction to matrices

A matrix is a rectangular arrangement of numbers in rows and columns, enclosed in brackets and treated as a single object. Biology and laboratory work produce data in exactly this form: colony counts for several plates at several times, absorbance readings for several samples at several wavelengths, or the numbers of organisms in each age group of a population. Writing such data as a matrix lets a technician add, scale, combine and solve for unknowns with a small set of rules instead of handling every number separately. Example: a technician counts the colonies that have grown on three nutrient agar plates after 24 hours and again after 48 hours. The six counts form one matrix with a row for each plate and a column for each time.

Plate   24 h   48 h
  1      12     31
  2      14     35
  3       9     27

Written as a matrix, the same data become:

      [ 12  31 ]
A  =  [ 14  35 ]
      [  9  27 ]

Rows, columns and entries

A row runs across the matrix and a column runs down it. In the matrix A above, the first row is 12 and 31, and the second column is 31, 35 and 27. Each number inside a matrix is an entry, also called an element. The entry in the first row and second column of A is 31, the 48-hour count of plate 1. Rows are always counted from the top and columns from the left, and the row number is always stated first.

Notation

A matrix is named with a capital letter such as A, B or C. Its entries are named with the matching small letter and two small numbers: a₁₂ means the entry in row 1, column 2. In general, aᵢⱼ is the entry in row i and column j. For the matrix A above:

• a₁₁ = 12, the 24-hour count of plate 1.
• a₂₂ = 35, the 48-hour count of plate 2.
• a₃₁ = 9, the 24-hour count of plate 3.

Brackets, either square [ ] or round ( ), are drawn round the entries so that the whole arrangement reads as one object. Bars | | mean something different (a determinant), so they are not used to write a matrix.

Equal matrices

Two matrices are equal only when they have the same number of rows, the same number of columns and every pair of matching entries is equal. If a second set of plates gave counts that differ in even one position, the two matrices are not equal. This rule is used later to turn a matrix equation into ordinary equations: matching entries are set equal one by one.

Uses of matrices in biology and laboratory work

• Storing data: A table of counts, masses or readings becomes a matrix that can be processed as a whole, for example when two batches of readings are added.
• Scaling readings: A single number can multiply every entry at once, for instance when counts from a small aliquot are converted to counts per millilitre.
• Solving unknowns: Two or more measured totals that depend on two or more unknown quantities can be written as one matrix equation and solved with an inverse matrix.
• Projecting populations: The number of organisms in each life stage can be stored in a column, and a matrix of survival and reproduction rates gives the numbers in the next generation.
• Combining tables: A table of reagent volumes and a table of reagent prices combine by matrix multiplication into the cost of each procedure.