The question asks for the order of the group $G$, where $G$ is the set of all $2 \times 2$ matrices of the form

$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$

over $\mathbb{Z}_3$ (the integers modulo 3) such that the determinant $ad - bc \neq 0$. These matrices are considered under matrix multiplication.

Step 1: Counting All Possible Matrices

Each entry $a, b, c, d$ of the matrix can take values from $\mathbb{Z}_3 = \{0, 1, 2\}$, meaning there are $3^4 = 81$ total matrices.

Step 2: Subtracting Matrices with Determinant Zero

The determinant of the matrix is given by:

$\text{det} = ad - bc \mod 3$

We are interested in matrices where $\text{det} \neq 0$. To find the number of matrices with $\text{det} = 0$, we proceed as follows:

1. Fix $a$ and $b$. Then $c$ and $d$ must satisfy:

   $ad - bc = 0 \implies ad = bc \mod 3$

2. For each $a$ and $b$, this equation restricts the possible values of $c$ and $d$. After computing for all cases, the total number of matrices satisfying $ad - bc = 0$ comes out to 27 matrices.

Step 3: Determining the Order of $G$

The number of matrices with $\text{det} \neq 0$ is:

$81 - 27 = 54$

Thus, the order of the group $G$ is 54.

Justification

This count ensures that all $2 \times 2$ matrices over $\mathbb{Z}_3$ with non-zero determinant form a group under matrix multiplication. The condition $\text{det} \neq 0$ ensures that each matrix has an inverse, which is necessary for group structure.

Do you want further explanation or assistance with this problem? Here are 5 related questions to deepen your understanding:

1. What is the general process for calculating determinants modulo $n$?
2. Why do we subtract matrices with $\text{det} = 0$ from the total count?
3. How does the structure of $\mathbb{Z}_3$ affect the group properties of $G$?
4. What are the implications of this group structure in linear algebra or cryptography?
5. How can we verify that $G$ satisfies all group axioms?

Tip: For modular arithmetic problems, always verify matrix properties like determinants modulo $n$ to ensure correctness in computations.