Overview:

Part 9: Determinants of 3 × 3 matrices

Minors and cofactors

Before we can find the determinant of a 3 × 3 (or larger) square matrix, we need to learn some new terminology.

Each element of a square matrix has a minor. The minor of the element is found by removing the row and column containing that element, and calculating the determinant of the remaining matrix.

Each element of a square matrix also has a cofactor. The cofactor of the element is either equal to its minor multiplied by 1, or its minor multiplied by -1. In other words, it is either the minor itself or the minor with its sign changed. We decide whether to keep or change the sign as follws: the cofactor of the element in the \(i\text{th}\) row and \(j\text{th}\) column is the minor of the element multiplied by \((-1)^{i+j}\). In other words, if \(i+j\) is even, we keep the minor’s sign the same, and if \(i+j\) is odd, we change its sign. For a 3 × 3 matrix, the following shows how each element’s cofactor is related to its minor:

\(\begin{pmatrix} \text{keep} & \text{change} & \text{keep}\\ \text{change} & \text{keep} & \text{change} \\ \text{keep} & \text{change} & \text{keep} \end{pmatrix}\)

This applet guides you through the process, step-by-step:

Finding the determinant

To find the determinant of a matrix, we need to find the cofactors of all of the elements in just one row or column This means that we can find the determinant of a 3 × 3 matrix without needing to find the cofactors of all nine elements.

To find the determinant, pick a row or column. For that row or column, multiply each element by its cofactor and note the product, and finally add these products.

Whichever row or column you choose, you should get the same answer, but it will save time if you choose a row or column with zeroes in it, if possible. If you have a 0 element, then when you multiply this by its cofactor, you will get 0. This means that you can save time by not bothering to find that element’s cofactor.

Use this applet to practise finding the determinant. To start with, it would be good to repeat each question using a different row or column to verify that you always get the same answer.

What the determinant represents

In part 5, we saw that the determinant of a 2 × 2 matrix M is equal to the area scale factor by which M transforms the areas of shapes. The determinant of a 3 × 3 matrix M is equal to the volume scale factor by which M transforms the volume of shapes. (We can also extend this idea to higher dimensions, though it is very hard to visualise object (let alone transformations of such objects) beyond the third dimension!)