Determinants

Martin McBride
9 min readNov 30, 2023

The determinant of a matrix is a scalar value (ie a single number) calculated from the elements of the matrix.

In this article, we will look at the definition and some of the uses of determinants. We will also look at some properties of determinants. The determinant is only defined for square matrices.

What are determinants used for?

We can use matrices to perform transformations (scaling, rotation, etc) in 2D or 3D spaces. They are used a lot in computer graphics, for example. The inverse of a matrix represents the inverse transform (for example scaling by 2 and scaling by 0.5 are inverses of each other). Determinants are used in the calculation of inverse matrices. The reason we need the determinant is that it represents the scale factor of a general special transform, as we will see below.

We can use matrices to solve linear simultaneous equations, and again the determinant is used in that calculation.

We also need to know the determinant to calculate the eigenvectors of a matrix. We will cover this in a later article.

Determinant of a 2 by 2 matrix

We will start by looking at how a determinant is calculated. For a 2 by 2 matrix like this:

We calculate the determinant like this:

The determinant is a single value formed from all 4 values in the original matrix. For example:

The determinant can also be written like this:

Determinant of a 3 by 3 matrix

Here is a 3 by 3 matrix:

We calculate the determinant like this:

This is called the Laplace expansion. What we have done here is to work along the top row of the…

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