Hi there. This post is about the linear algebra topic of computing determinants of 2 by 2 and 3 by 3 matrices. This post will go over the formulas and not through the cofactor method. The cofactor method in computing determinants will be in a different post.
The determinant of a matrix is a number. On its own it does not have much application but it does influence a lot of results in the mathematical field of linear algebra.
The notation for the determinant of a matrix Ais often denoted by . Sometimes instead of having the square brackets like these [ ], you would have vertical bars (like absolute values) denoting the determinant of a matrix. Here is a 3 by3 determinant example:
Computing 2 by 2 and 3 by 3 Determinants
There is an alternate and easier way of computing determinants for 2 by 2 and 3 by 3 matrices. We have formulas and memory aids to compute such determinants. These do not work for higher dimensional matrices such as 4 by 4 or a 1000 by 1000 matrix.
Two by Two Determinant Case
Suppose the 2 by 2 square matrix A is of the form:
The determinant of A is .
Given a 2 by 2 square matrix A of the form as above and the determinant is non-zero then the inverse of matrix A is:
The entries and are switched and entries and switch signs. In addition, you see why the determinant has to be non-zero. The math police will be after you if you divide by zero!
Three by Three Determinant Case
The formula for computing the determinant of a 3 by 3 matrix is more involved than the 2 by 2 case but it is not too hard once you understand it.
Suppose the matrix A is of the form:
To compute the 3 by 3 determinant, we add another a,b,c column and add another d,e,f column to the right side of A. A picture below will help illustrate this.
The “Tricky” Part
It is somewhat long to explain in words. If you would like a visual (and summary) of the below steps, please refer to the image below.
Starting from the top left entry we draw a downward right diagonal line to get one of six terms which is . Then from entry in row 1, column 2 draw another downward right diagonal line to get the term . Continue from in row 1, column 3 and to the same to obtain .
(Recall that row 1 is the top row of the matrix and column 1 is the most left vertical column.)
So far we have .
To obtain the last three terms we do a similar procedure but starting from the top right and we go to the downward left direction. Also, these next three terms out of six are negative.
Starting from top right , make a downward diagonal left line to get the term . From the second in the fourth column, we get and the last term gets us .
From these three terms we have .
The determinant of the matrix A for a 3 by 3 matrix is:
I don’t know about you but this looks nasty to memorize. It is better to memorize the steps in obtaining this determinant than the formula.
I have included an image illustrating the method and formula for the 3 by 3 determinant.
These examples will show how the formulas are used in computing determinants for 2 by 2 and 3 by 3 matrices.
The inverse of a 3 by 3 matrix does not involve its determinant.
These images are from my own camera phone.