Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{cc} x & x \ln x \\ 1 & 1+\ln x \end{array}\right|$$

Step-by-Step Solution

Answer

The result of the determinant is \(x\).

Step 1: Identify the elements of the matrix

The identifier matrix contains 4 elements: \(a = x\), \(b = x \ln x\), \(c = 1\), and \(d = 1+\ln x\).

Step 2: Apply the formula for the determinant of a 2x2 matrix

Substitute the identified elements into the formula: \( (x * (1+\ln x)) - ((x \ln x) * 1) \)

Step 3: Solve the equation

Calculate the equation: \(x + x \ln x - x \ln x = x\). Once you simplify the result, you will find that the equation merely gives \(x\).

Key Concepts

2x2 Matrix

A 2x2 matrix is a grid of numbers arranged in two rows and two columns. It is the simplest form of a square matrix, which means all columns and rows have equal length. In our example, the matrix is:
\[\begin{pmatrix} x & x \ln x \ 1 & 1+\ln x \end{pmatrix}\]This matrix has entries that are functions of the variable \(x\), which makes it interesting when applying calculus concepts.
To evaluate a 2x2 matrix's determinant, we use a specific formula: for a matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is calculated as \(ad - bc\). This formula is a cornerstone in linear algebra that informs us about the area scaling factor when the matrix is applied as a linear transformation.

Function Entries

In the matrix provided, the entries are not just numbers, but functions of a variable, \(x\). This is notable because it adds a layer of complexity to the process of finding the determinant.
The entry "\(x\)" is straightforward, but "\(x \ln x\)" and "\(1 + \ln x\)" introduce logarithmic functions, which require careful manipulation. Logarithmic functions, like \(\ln x\), often appear in calculus because they have unique derivative properties that simplify many equations.
Functions as matrix entries are common in advanced applications, such as solving differential equations or performing variable changes in calculus. Understanding how each function changes with different values of \(x\) helps in grasping how the matrix transforms in different scenarios.

Variable Changes

In calculus, changing variables is a common practice to simplify complex equations or integrals. When we change variables, we adjust the framework we use to examine a problem, often making difficult calculus operations more manageable.
Our matrix example reflects how determinants are used when variables change. The determinant \(x + x \ln x - x \ln x = x\) simplifies to just \(x\), showing how the transformation affects the representation of the function.

• First, you recognize functions as matrix elements.
• Next, apply the determinant formula to find how each element interacts.
• Finally, simplify the result to see the impact of variable changes.

Changes of variable are often employed in integral transformations, such as moving from Cartesian to polar coordinates, where determinants play a crucial role in determining the scaling factor.

Calculus

Calculus, the study of continuous change, is a vast field that often uses matrices and determinants, especially in scenarios involving transformations or optimization problems.
Derivatives and integrals, two primary operations in calculus, respond dynamically to transformations captured by matrix operations. For example, the calculation of a Jacobian matrix—a matrix of all first-order partial derivatives of a vector-valued function—is rooted in determinant theory.
In the context of our example, understanding calculus helps you see beyond the numbers. You learn how transformations represented by determinants provide insights into changes in scale or dimension among multiple variables.
Recognizing the connection between determinants and transformations in calculus allows us to tackle complex multivariable problems with ease. Whether you're performing integrations or exploring dynamic systems, calculus and determinants enable precise manipulation and understanding of mathematical landscapes.