2 x 2 and 3 x 3 Determinants

# 2 x 2 and 3 x 3 Determinants

We are about to look at The Cross Product of Vectors in Three-Dimensional Space, but before we do, we will first look at an important object known as determinants that will help us in evaluating the cross product between vectors in $\mathbb{R}^3$.

Determinants are a property of objects known as Square Matrices which are arrays of numbers. A matrix with $2$ rows and $2$ columns has size $2 \times 2$ and has four entries. Matrices of size $2 \times 2$ look like $\begin{bmatrix}a & b\\ c & d \end{bmatrix}$. A matrix with $3$ rows and $3$ columns has size $3 \times 3$ and has nine entries. Matrices of size $3 \times 3$ look like $\begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i \end{bmatrix}$. We will not go too far into matrices and their many uses and important properties, but if the reader is interested, they are available to be read on the Linear Algebra section of this site.

 Definition: The Determinant of a $2 \times 2$ Matrix $\begin{bmatrix}a & b\\ c & d \end{bmatrix}$ denoted $\begin{vmatrix}a & b\\ c & d \end{vmatrix} = ad - bc$. The Determinant of a $3 \times 3$ Matrix $\begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i \end{bmatrix}$ is $\begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix} = a \begin{vmatrix}e & f\\ h& i \end{vmatrix} - b \begin{vmatrix}d & f\\ g& i \end{vmatrix} + c \begin{vmatrix}d & e\\ g& h \end{vmatrix}$.

We should also note that the determinant of a $3 \times 3$ matrix can be calculated in many different ways, however, the formula provided above will be the only one necessary for formulating the cross product between vectors subsequently. We will now look at some important properties regarding determinants.

 Theorem 1: For any square matrix $A$, the determinant of $A$ has the following properties: a) If two rows of the determinant of $A$ are equal, then the determinant will equal zero. b) If two rows of the determinant of $A$ are interchanged, then the determinant changes signs. c) If a multiple of one row is added or subtracted to another row, then the determinant is unchanged.

We will not prove any parts to Theorem 1 as they require some Linear Algebra. However, we will now look at some examples of calculating determinants.

## Example 1

Calculate the determinant of the matrix $\begin{bmatrix}1 & 4\\ 5 & 5 \end{bmatrix}$.

Applying the formula for $2 \times 2$ determinants and we get that:

(1)
\begin{align} \begin{vmatrix}1 & 4\\ 5 & 5 \end{vmatrix} = (1)(5) - (4)(5) = 5 - 20 = -15 \end{align}

## Example 2

Calculate the determinant of the matrix $\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 1 & 2 & 3 \end{bmatrix}$.

By Theorem 1 part (a) since the first and third row are equal then the determinant of this matrix is $0$. This can also be shown by applying the formula directly to this matrix.

## Example 3

Calculate the determinant of the matrix $\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}$.

Applying the formula for $3 \times 3$ determinants and we get that:

(2)
\begin{align} \quad \quad \begin{vmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{vmatrix} = 1 \begin{vmatrix} 5 & 6\\ 8 & 9 \end{vmatrix} - 2 \begin{vmatrix} 4 & 6\\ 7 & 9 \end{vmatrix} + 3\begin{vmatrix} 4 & 5\\ 7 & 8 \end{vmatrix} = 1(45 - 48) - 2(36 - 42) + 3(32 - 35) = -3 +12 - 9 = 0 \quad \end{align}