Determinants in Linear Algebra

A determinant is a numerical value attached to a square matrix. Whether this determinant is 0 or not, is one of the key properties of any square matrix.

Intuition for determinants in 2D space

In the last section, you understood that a matrix represents a linear transformation. You also saw that this linear transformation also distorts the n-dimensional space. What does a determinant mean, in this sense of space-distortion?

Intuitively, in 2D space, the determinant of a matrix (i.e., a linear transformation) is the area that 1 square unit is expanded/shrunk to, when the linear transformation expressed by the matrix is applied on 2D space.

Let's say we have the following 2x2 matrix A:

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

The determinant is given by the formula:

$det\left(A\right)=a\ast d-b\ast c$

Thus, in the distorted space, 1 square unit of area is multiplied by a factor of $(a\ast d-b\ast c)$ when matrix A is applied as a linear transformation.

How does this scale to 3x3 matrices?

$B=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$

The determinant is given by:

$det\left(B\right)=a\ast det\left(\left[\begin{array}{cc}e& f\\ h& i\end{array}\right]\right)-b\ast det\left(\left[\begin{array}{cc}d& f\\ g& i\end{array}\right]\right)+c\ast det\left(\left[\begin{array}{cc}d& e\\ g& h\end{array}\right]\right)$

For 3D space, the determinant represents distortion of volume, instead of area.

This process of calculating determinants replicates itself exactly for matrices of higher dimensions.

This type of determinant calculation will appear in the section on eigenvalues and eigenvectors.

Determinants and Inverses

The determinant of a matrix is closely linked with its inverse. Specifically, a matrix is singular (non-invertible) if and only if its determinant is equal to zero. Also, both of these properties (singular and determinant is 0) will hold true for any matrix whose columns are linearly dependent.

Today, your computer package will perform the actual calculations of an inverse within seconds. While working with matrices, though, it is beneficial to run quick checks on determinants to ensure you are not ending up with singular matrices. If you're interested in the exact calculations of the inverse, we have provided links to videos in the Additional Reading section.

Additional Reading

1. Here is a youtube video that elegantly describes determinants, and their relationship with 2D & 3D space
2. Part 1 of calculating inverse of a matrix using the determinant
3. Part 2 of calculating inverse of a matrix using the determinant