Percentage changes with a diagonal matrix (application)

A diagonal matrix of multipliers

A diagonal matrix is a square matrix whose entries off the leading diagonal (top left to bottom right) are all 0, for example \(\begin{pmatrix}1.1&0\\ 0&0.8\end{pmatrix}\).

Each entry on the diagonal is a multiplier. An increase of \(r\%\) multiplies by \(1+\frac{r}{100}\), a decrease of \(r\%\) multiplies by \(1-\frac{r}{100}\), and "decreases to 80%" multiplies by \(0.8\). A profit of 30% of a price is \(0.3\) times the price.

Why each row or column gets its own factor

\(\begin{pmatrix}a&0\\ 0&b\end{pmatrix}\begin{pmatrix}p&q\\ r&s\end{pmatrix}=\begin{pmatrix}ap&aq\\ br&bs\end{pmatrix}\) On the left: row 1 is multiplied by \(a\) and row 2 by \(b\).

\(\begin{pmatrix}p&q\\ r&s\end{pmatrix}\begin{pmatrix}a&0\\ 0&b\end{pmatrix}=\begin{pmatrix}ap&bq\\ ar&bs\end{pmatrix}\) On the right: column 1 is multiplied by \(a\) and column 2 by \(b\).

A diagonal matrix times a column matrix multiplies each entry of the column by its own factor: \(\begin{pmatrix}a&0\\ 0&b\end{pmatrix}\begin{pmatrix}p\\ q\end{pmatrix}=\begin{pmatrix}ap\\ bq\end{pmatrix}\). With three matrices, such as \(QRP\), multiply in two steps: \(Q(RP)\) gives the same answer as \((QR)P\), so you may work out \(RP\) first.

In the row-by-column rule the zeros remove every product except one, so each entry is multiplied by one factor only. A single percentage for every entry needs only a scalar, such as \(1.07A\); a diagonal matrix is needed when the categories change by different percentages.

Method

1. Decide which categories change, and find the multiplier for each.
2. Write the multipliers on the leading diagonal in the same order as those categories, with 0 everywhere else.
3. Put the diagonal matrix on the left if the categories are the rows, on the right if they are the columns. Check that the orders allow the product.
4. Multiply, and state the meaning and unit of each entry.

Example

A museum sells adult and child tickets. Next weekend, adult sales are expected to rise by 10% and child sales to fall by 20%.

Adult and child are the columns, so multiply on the right by \(D=\begin{pmatrix}1.1&0\\ 0&0.8\end{pmatrix}\). The orders \((2\times2)(2\times2)\) give a \(2\times2\) answer.

\(TD=\begin{pmatrix}120(1.1)&80(0.8)\\ 150(1.1)&95(0.8)\end{pmatrix}=\begin{pmatrix}132&64\\ 165&76\end{pmatrix}\)

Putting \(D\) on the left instead gives \(DT=\begin{pmatrix}132&88\\ 120&76\end{pmatrix}\), which raises Saturday by 10% and lowers Sunday by 20%: the wrong categories.

Check

A decrease of 20% uses the multiplier \(0.8\), not \(0.2\) or \(-0.2\). Check one entry by ordinary arithmetic, for example \(120\times1.1=132\).