Problem 92 - Precalculus

Question

In calculus, determinants are used when evaluating double and triple integrals through a change of variables. In these cases, the elements of the determinant are functions. Find each determinant. $$\left|\begin{array}{cc} 2 x & 2 y \\ 2 x & 2 y-2 \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant is \(-4x\).

Step 1: Identify the Matrix

The given determinant is a 2x2 matrix:\[ \begin{vmatrix} 2x & 2y \ 2x & 2y - 2 \ \end{vmatrix} \]

Step 2: Use Formula for 2x2 Determinant

The formula for calculating the determinant of a 2x2 matrix \( \begin{vmatrix} a & b \ c & d \end{vmatrix} \) is given by: \[ ad - bc \] In this problem, \(a = 2x\), \(b = 2y\), \(c = 2x\), and \(d = 2y - 2\).

Step 3: Substitute the Elements into the Formula

Substitute the values found in Step 2 into the determinant formula:\[ (2x)(2y-2) - (2y)(2x) \]

Step 4: Simplify the Expression

Simplify each term from Step 3:1. Calculate \((2x)(2y-2) = 4xy - 4x\).2. Calculate \((2y)(2x) = 4xy\).3. Subtract the second term from the first: \[ 4xy - 4x - 4xy \] The \(4xy\) terms cancel out, and we are left with \[ - 4x \].

Step 5: Write the Final Answer

The determinant is equal to:\[ -4x \]

Key Concepts

2x2 matrix

A 2x2 matrix is a simple but essential element in mathematics, especially in linear algebra. It consists of two rows and two columns, which makes it a square matrix. Each element within this arrangement can be a number or even a function, depending on the context. When dealing with calculus, these elements might be expressions involving variables like the matrix we have here. For example:

• First row:
  • First element: 2x
  • Second element: 2y
• Second row:
  • First element: 2x
  • Second element: 2y - 2

The arrangement of these elements makes it possible to perform various operations, such as addition, multiplication, and finding determinants.

determinant calculation

The calculation of a determinant is crucial in various fields of mathematics, including calculus. For a 2x2 matrix, the determinant helps reveal specific properties related to linear transformations and areas.
The determinant of a 2x2 matrix \(\begin{vmatrix} a & b \ c & d \end{vmatrix}\) can be calculated using the formula:

• \(ad - bc\)

Here's how it works in our context:

• \(a = 2x\)
• \(b = 2y\)
• \(c = 2x\)
• \(d = 2y - 2\)

By substituting into the formula, it becomes:

• \((2x)(2y-2) - (2y)(2x) = 4xy - 4x - 4xy\)

After simplifying, the result is \(-4x\). This step-by-step method ensures accuracy by methodically breaking down each part of the calculation.

integrals and determinants

In calculus, determinants play a pivotal role, especially when dealing with integrals. While integrals measure areas and volumes, determinants simplify transformations in these computations.
When working with double or triple integrals, changing variables is often necessary. This transformation involves determinants because they provide a scaling factor that relates to the area or volume change during the transformation process.
A clear understanding of the determinant helps evaluate integrals more effectively, especially when the areas and volumes need to be determined after a variable change. This application makes determinants a powerful tool in calculus, linking algebraic concepts tightly with integral calculus.
Therefore, mastering the calculation of determinants directly contributes to comprehending more advanced calculus concepts and ensures a smoother transition from algebra to calculus applications.