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Use row and column matrices of ones to extract totals

Start a set of 8
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Sales F, rows Rose and Tulip, columns Fri, Sat and Sun
\(\displaystyle F=\begin{bmatrix}20&35&15\\12&18&24\end{bmatrix}\)

A florist records bunches sold in the matrix F. Which product gives the number of each flower sold over the three days?

Year 12, VCE Units 3 & 4, Chapter 10, 4 difficulty levels, No calculatorBuild a worksheet

Extract row and column totals

Use F with rows [20, 35, 15] and [12, 18, 24], for Rose and Tulip sales over Fri, Sat and Sun. A column of ones on the right gives flower totals, while a row of ones on the left gives day totals

\[F\begin{bmatrix}1\\1\\1\end{bmatrix}=\text{row totals}\]
  1. Put a column of ones on the right Right multiplication by a column of ones adds across each row \(\displaystyle F\begin{bmatrix}1\\1\\1\end{bmatrix}=\begin{bmatrix}70\\54\end{bmatrix}\)
  2. Put a row of ones on the left Left multiplication by a row of ones adds down each column \(\displaystyle \begin{bmatrix}1&1\end{bmatrix}F=\begin{bmatrix}32&53&39\end{bmatrix}\)
  3. Match the inner dimensions The adjacent dimensions must be equal for the product to be defined \(\displaystyle (2\times3)(3\times1)\text{ is defined}\)
  4. Use both sides for a grand total A row of ones, the data matrix and a column of ones produce one grand total \(\displaystyle (1\times2)(2\times3)(3\times1)=1\times1\)

Worked example

Let A have rows [3, 8, 4] and [6, 2, 5]. Use a column of ones to find both row totals

\[A=\begin{bmatrix}3&8&4\\6&2&5\end{bmatrix},\quad J=\begin{bmatrix}1\\1\\1\end{bmatrix}\]
  1. \(\displaystyle J=\begin{bmatrix}1\\1\\1\end{bmatrix}\text{ has order }3\times1\)The column of ones has three entries, matching the three columns of A
  2. \(\displaystyle 3\times1+8\times1+4\times1=15\)Multiplying the first row by J gives the first row total of 15
  3. \(\displaystyle 6\times1+2\times1+5\times1=13\)Multiplying the second row by J gives the second row total of 13
  4. \(\displaystyle AJ=\begin{bmatrix}15\\13\end{bmatrix}\)The product lists the two row totals in the same row order as A

Common mistakes

Finding day totals instead of flower totals
✕\(\displaystyle \begin{bmatrix}1&1\end{bmatrix}F=\begin{bmatrix}32&53&39\end{bmatrix}\)✓\(\displaystyle F\begin{bmatrix}1\\1\\1\end{bmatrix}=\begin{bmatrix}70\\54\end{bmatrix}\)
A column of ones on the right adds each flower row, while a row of ones on the left adds each day column
Using the wrong ones-matrix shape
✕\(\displaystyle A\begin{bmatrix}1&1\end{bmatrix}\)✓\(\displaystyle A\begin{bmatrix}1\\1\\1\end{bmatrix}\)
For a 2 by 3 data matrix, the right-hand factor must have three rows so the inner dimensions match
Using entrywise multiplication
✕\(\displaystyle F\circ\begin{bmatrix}1&1&1\\1&1&1\end{bmatrix}=F\)✓\(\displaystyle F\begin{bmatrix}1\\1\\1\end{bmatrix}=\begin{bmatrix}70\\54\end{bmatrix}\)
Matrix multiplication forms row-by-column sums, whereas entrywise multiplication leaves the data entries unchanged here

Difficulty levels

introUse a supplied column or row of ones to find row or column totals
standardConstruct the required ones matrix and use it on the correct side
challengeFind both axis totals, a grand total or a mixed product with a value column
examChoose the correct side, shape or product from alternatives with multiplication errors

Questions about this topic

Why does a column of ones on the right give row totals?+

Each row is multiplied by every entry in the column of ones, so the row entries are added together

What order should the column of ones have?+

It must have one row for each column of the data matrix. For a 2 by 3 matrix, use a 3 by 1 column

How can a grand total be found by matrix multiplication?+

Multiply a row of ones on the left, the data matrix, and a column of ones on the right. The resulting 1 by 1 matrix is the grand total

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, General Mathematics, chapter 10: Matrices

In this chapter

State the order of a matrix and locate its elementsIdentify row, column, square, symmetric and zero matricesRead information from a labelled data matrixRead, transpose and classify a matrixConstruct a matrix from labelled information or an entry rule10A Matrix notation, types and construction: Solve a matrix equation10A Matrix notation, types and construction: Construct, apply and invert a permutation matrixAdd and subtract matricesFind unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matrices10B Linear operations on matrices: Determine whether an inverse exists and calculate itDecide whether a matrix product is defined and give its orderMultiply matricesEvaluate powers of a square matrixIdentify and construct the identity matrix10C Matrix products, powers and permutations: Construct, apply and invert a permutation matrixFind 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 matrix10D Determinants, inverses and matrix equations: Determine whether an inverse exists and calculate it10D Determinants, inverses and matrix equations: Solve a matrix equationUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totalsRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixConstruct and analyse relation matrices