Determinants Of 2 By 2 And 3 By 3 Matrices

Introduction

 

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 \(\text{det}(A)\). 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:

 

\[\displaystyle \text{A} = \begin{vmatrix} a & d & g \\ b & e & h\\ c & f & i \\ \end{vmatrix} \]

 

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:

 

\[\displaystyle \text{A} = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \]

 

The determinant of A is \(\text{det}(A) = ad - bc\).

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:

 

\[\displaystyle \text{A}^{-1} = \dfrac{1}{\text{det}(A)}\begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix} \, = \, \dfrac{1}{(ad - bc)}\begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix} \]

 

The entries \(a\) and \(d\) are switched and entries \(b\) and \(c\) 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:

\[\displaystyle \text{A} = \begin{bmatrix} a & b & c \\ d & e & f\\ e & f & g \\ \end{bmatrix} \]

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.

 

Image Source

 

Image Link Source: http://thejuniverse.org/PUBLIC/LinearAlgebra/LOLA/detDef/special.html

 

Starting from the top left \(a\) entry we draw a downward right diagonal line to get one of six terms which is \(aei\). Then from entry \(b\) in row 1, column 2 draw another downward right diagonal line to get the term \(bfg\). Continue from \(c\) in row 1, column 3 and to the same to obtain \(cdh\).

(Recall that row 1 is the top row of the matrix and column 1 is the most left vertical column.)

So far we have \(aei + bfg + cdh\).

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 \(c\), make a downward diagonal left line to get the term \(ceg\). From the second \(a\) in the fourth column, we get \(afh\) and the last term gets us \(bdi\).

From these three terms we have \(-ceg - afh - bdi\).

The determinant of the matrix A for a 3 by 3 matrix is:

 

\[\displaystyle \text{det}(A) = [aei + bfg + cdh] - [ceg + afh + bdi]\].

 

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.