Problem 89 - 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. \(\left| \begin{array}{r} x & \textrm{ln}\ x \\ 1 & 1/x \end{array} \right|\)

Step-by-Step Solution

Answer

The determinant of the given 2x2 matrix is 1 - ln(x)

Step 1: Identify the elements in the matrix

Looking at the given 2x2 matrix \( \left| \begin{array}{cc} a & b \ c & d \end{array} \right| \), where a=x, b=ln(x), c=1 and d=1/x.

Step 2: Apply the formula for the determinant

Using the formula for the determinant of a 2x2 matrix \( \text{det}(A) = ad - bc \), the determinant for the given matrix is \( \text{det}(A) = (x * 1/x ) - (\text{ln}(x) * 1) = 1 - \text{ln}(x) \)

Step 3: Simplify the solution

Simplification is not required in this case, because the expression cannot be reduced further. Thus, the determinant of the given matrix is 1 - ln(x)

Key Concepts

Matrix Algebra

Matrix algebra involves the study and manipulation of matrices. A matrix is a rectangular array of numbers, symbols, or expressions, organized in rows and columns. In this context, understanding matrices is crucial because they are widely used in various fields such as physics, computer science, and engineering. However, some operations are unique and specific to matrix algebra:

• Addition and Subtraction: Matrices can be added or subtracted only if they have the same dimensions.
• Scalar Multiplication: Each element in the matrix is multiplied by a scalar value.
• Multiplication: Matrices can be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix.
• Determinant: The determinant is a scalar value that can be derived from a square matrix, providing significant insights into the matrix's properties.

Understanding these operations allows for deeper comprehension of linear transformations and systems of linear equations.

Calculus

Calculus is a branch of mathematics focusing on changes, and it can be divided mainly into differential and integral calculus. In relation to matrices, calculus often involves functions that can be expressed using matrices. Determinants arise naturally when considering transformations and changes of variables in calculus.
Here are a few key points to consider:

• Derivatives: Help determine the rate of change of a function, which is useful in optimization problems or when changing variables.
• Integrals: Used to calculate areas under curves and are essential for solving differential equations.
• Determinants in Calculus: They are used to evaluate Jacobians in multiple integrals or transformations.

Thus, mastering how determinants play a role in calculus will help solve complex problems involving changes of variables.

Change of Variables

The method of change of variables is a powerful tool in calculus. It simplifies the process of evaluating integrals and solving differential equations by transforming complex expressions into simpler forms.
This concept is especially useful when dealing with functions of multiple variables:

• Jacobian Matrix: When changing variables in multiple integrals, the Jacobian determinant is crucial. It modifies the scale of integration according to the variables' transformation.
• Simplification: By choosing suitable variables, calculations become simpler and more manageable.
• Applicability: Useful in physics for describing coordinate transformations.

Employing the change of variables effectively requires familiarity with how matrices and their determinants function.

Determinant Formula

The determinant formula for a 2x2 matrix helps calculate its determinant, a quantity indicating certain properties of the matrix such as invertibility or volume scaling in a transformation.
The formula is straightforward: for a matrix \[\begin{array}{cc}a & b \c & d\end{array}\], the determinant is computed as:

• Formula: \( ext{det}(A) = ad - bc \)

Key things to note about determinants include:

• Zero Determinant: Implies the matrix is not invertible.
• Non-zero Determinant: Indicates an invertible matrix.
• Practical Usage: Can be used to find area and volume affected by the matrix transformations.

Determinants are central in many mathematical applications, making it vital to understand and apply the determinant formula correctly.