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In mathematics, the determinant of a matrix is a special number associated with it which provides useful properties about the matrix, especially when dealing with linear equations and transformations. For a 2x2 matrix, which is the simplest type of matrix containing two rows and two columns, calculating the determinant is straightforward.
The 2x2 matrix is typically written as:

• \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \)

The determinant of this matrix, denoted as \( \text{det} \), is given by the expression:

• \( \text{det} = ad - bc \)

This formula considers the diagonal elements of the matrix and computes the difference between the product of the diagonal elements from the products of the off-diagonal elements. This key formula is foundational for studying matrices in linear algebra.

Modular arithmetic is a system of arithmetic for integers where numbers "wrap around" upon reaching a certain value, known as the modulus. This concept is integral to many areas of mathematics and computer science. It is often illustrated with a clock, where the numbers reset to zero after reaching a certain point.
For instance, in modular arithmetic with modulus 7, which is denoted as \( \mathbb{Z}_7 \), we consider the numbers from 0 to 6. Any integer calculation done in \( \mathbb{Z}_7 \) requires reducing the result to a number within these bounds:

• \( a \mod 7 \)

This means you take \( a \) (the result of your arithmetic operation) and find the remainder when \( a \) is divided by 7. For example, \( 10 \mod 7 = 3 \) because dividing 10 by 7 leaves a remainder of 3.

The matrix determinant formula is a critical tool in linear algebra, allowing for the evaluation of the determinant for various sizes of matrices. For the 2x2 matrix we are focusing on, the formula \( \text{det} = ad - bc \) is particularly easy to apply and compute.
Here's how it works step by step:

• Identify the elements in the matrix. For \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), \( a \), \( b \), \( c \), and \( d \) are the matrix components.
• Multiply \( a \) and \( d \), the main diagonal elements.
• Multiply \( b \) and \( c \), the off-diagonal elements.
• Subtract the product of the off-diagonal from the product of the main diagonal.

In practice, for our matrix \( \begin{pmatrix} 4 & 2 \ 5 & 3 \end{pmatrix} \), we compute:

• \( 4 \times 3 = 12 \)
• \( 2 \times 5 = 10 \)
• Finally, \( 12 - 10 = 2 \) is the determinant.

Calculating determinants in \( \mathbb{Z}_7 \) follows the same determinant formula initially but requires applying modular arithmetic to the result. This requires every calculation to respect the modulus, such as recalculating numbers within 0 through 6.
For the matrix \( \begin{pmatrix} 4 & 2 \ 5 & 3 \end{pmatrix} \), follow these steps:

• Initially, calculate using the determinant formula: \( 4 \times 3 - 2 \times 5 = 12 - 10 = 2 \)
• Next, take the result through the modular operation, \( 2 \mod 7 = 2 \), since 2 is within our modular system \( \mathbb{Z}_7 \).

Thus, the determinant of the matrix in \( \mathbb{Z}_7 \) remains 2. This ensures our calculations are consistent with the modulus, reflecting a key principle of modular arithmetic where computations and results are constrained within a specific range.