Problem 85 - Precalculus with Limits

Question

In Exercises 85-90, evaluate the determinant in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. \(\left| \begin{array}{c} 4u & -1 \\ -1 & 2v \end{array} \right|\)

Step-by-Step Solution

Answer

The determinant of the matrix is \( 8uv - 1 \)

Step 1: Identify the elements of the matrix

The given 2x2 matrix is \[ \begin{array}{cc} 4u & -1 \ -1 & 2v \end{array} \]. In this matrix, a=4u, b=-1, c=-1 and d=2v.

Step 2: Apply the formula for the determinant

The determinant of the matrix is computed as ad - bc. Substituting our values, we obtain the expression: (4u*2v) - ((-1)*(-1)).

Step 3: Calculate the determinant

Performing the multiplication and subtraction, we find that the determinant of the matrix is 8uv - 1.

Key Concepts

2x2 Matrix

A 2x2 matrix is a basic building block in the study of linear algebra and calculus. It consists of two rows and two columns, resulting in four elements total. Understanding a 2x2 matrix is the foundation for many more complex matrix operations and concepts.

For example, a typical 2x2 matrix looks like this:

• The first row has the elements 'a' and 'b'.
• The second row has the elements 'c' and 'd'.

This can be represented mathematically as:\[\begin{array}{cc} a & b \ c & d \\end{array}\]A matrix can represent a system of linear equations, and its determinant, especially in 2x2 matrices, gives us valuable information about the system. If the determinant is zero, the system has no unique solutions. If not, then unique solutions exist.

Change of Variables

The change of variables is an essential concept in calculus and linear algebra. It involves transforming one set of variables into another, often to simplify a complex problem.

With matrices, especially those involving determinants, change of variables can drastically alter how the matrix behaves. A determinant with variables, like the example given, belongs to a scenario when we perform these transformations.

In the example given, the 2x2 matrix uses variables 'u' and 'v'. By treating these entries as functions, we essentially allow the matrix to represent more dynamic systems which vary with 'u' and 'v'. This property gives us the power to analyze how these variables transform behaviors or properties when altered. Not only does this help solve systems of equations, but it also assists in integration methodology where change of variables can simplify complex regions under study.

Multiplication and Subtraction in Matrices

Understanding how to multiply and subtract elements in matrices is crucial for determining their determinant, especially in 2x2 matrices. The determinant gives us insights into the matrix's properties, like if it is invertible or not.

For a 2x2 matrix, the determinant is found using the formula: \[ad - bc\]where 'a', 'b', 'c', and 'd' are the elements of the matrix. This straightforward operation involves:

• Multiplying the diagonal elements 'a' and 'd'.
• Multiplying the off-diagonal elements 'b' and 'c'.
• Subtracting the result of the off-diagonal multiplication from the diagonal multiplication.

For our example matrix with elements 4u, -1, -1, and 2v, the determinant is calculated as:\[(4u \times 2v) - ((-1) \times (-1)) = 8uv - 1\]This mathematical operation showcases how multiplication and subtraction combine to reveal essential insights about the matrix, allowing us to tackle more complex questions in calculus and linear systems.