Matrices Part 1: What is a Matrix?

computer screen with a graph of a financial data trend

Photo by M. B. M. on Unsplash

For economists and people working in finance, following currency exchange rates is crucial. This is one way to compare the relative strength of economies around the world. But there is a lot of data to keep track of!

One way to organize currencies so that exchange rates can quickly be compared is by arranging them in a rectangular array, or matrix. Here is an example of cross rates, as per Bloomberg, of some of the world’s largest economies on May 23, 2019:

\( \left[
\begin{array}{cccccc}
– & \text{USD} & \text{EUR} & \text{JPY} & \text{GBP} & \text{HKD} \\
\text{USD} & 1 & 1.1179 & 0.0091 & 1.2655 & 0.1274 \\
\text{EUR} & 0.8945 & 1 & 0.0082 & 1.132 & 0.114 \\
\text{JPY} & 109.57 & 122.51 & 1 & 138.672 & 13.9594 \\
\text{GBP} & 0.7901 & 0.8835 & 0.0072 & 1 & 0.1007 \\
\text{HKD} & 7.8492 & 8.7762 & 0.0716 & 9.9339 & 1 \\
\end{array}
\right] \)

(To see today’s cross rates, head over to Bloomberg)

A matrix is a table/array of elements arranged in horizontal rows and vertical columns. Elements in the same row/column share a particular property. The dimensions of a matrix are written as # of rows x # of columns. For example, a matrix with 2 rows and 2 columns is described as a 2×2 matrix (two-by-two matrix). You can read a matrix both sideways, from left to right, and vertically, from top to bottom. (You might also have to read some graphs this way on the Science section of the ACT.)

In the cross rate matrix above, each horizontal row and vertical column compares one particular currency to all of the others in the table. Each currency, by definition, is worth 1 of itself. For example, let’s look at column 1, row 1. 1 USD (US Dollar) = 1 USD. The non-1 numbers in column 1 tell you how much of another currency 1 USD can be exchanged for. For example, 1 USD = 109.57 JPY (Japanese Yen). The non-1 entries in row 1 tell you how many USD 1 of the other currencies are worth. For example, 1 GBP (Great British Pound) = 1.2655 USD.

Matrices are also commonly used in physics. They can be used to conveniently represent properties of physical objects, the polarization of light, and even model the behavior of space-time!

The entries in a matrix can represent a variety of things. For example, they can represent ratios and their inverses, like in the cross rate matrix above. They can also represent linear expressions or equations. Take, for example, the following system of linear equations:

\(2 x+y+z=10 \)

\(x+4 y+3 z=30 \)

\(-x+2 y+\frac{z}{2}=5 \)

A coefficient matrix is an array of the coefficients of the variables on the left-hand side of the equations:

\( \left[
\begin{array}{ccc}
2 & 1 & 1 \\
1 & 4 & 3 \\
-1 & 2 & \frac{1}{2} \\
\end{array}
\right] \)

Each row represents an equation, and each column represents a variable.

An augmented matrix takes the full equation and replaces the equals signs with bars:

\(\left[
\begin{array}{ccccc}
2 & 1 & 1 & | & 10 \\
1 & 4 & 3 & | & 30 \\
-1 & 2 & \frac{1}{2} & | & 5 \\
\end{array}
\right]\)

Now let’s look at what you can do with matrices.