Problem 88 - 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. \(\left| \begin{array}{c} e^{-x} & xe^{-x} \\ -e^{-x} & (1-x)e^{-x} \end{array} \right|\)

Step-by-Step Solution

Answer

The determinant of the given matrix is \(e^{-2x} - xe^{-2x} - x\).

Step 1 Title

Identify the elements of the 2x2 matrix: \(a=e^{-x}\), \(b=xe^{-x}\), \(c=-e^{-x}\) and \(d=(1-x)e^{-x}\).

Step 2 Title

Use the formula of determinant for a 2x2 matrix, which is given by \(ad-bc\). Thus, the determinant of this matrix is \(e^{-x}(1-x)e^{-x} - xe^{-x}(-e^{-x})\).

Step 3 Title

Calculate the determinant by performing the operations. This gives \(e^{-2x}(1-x) - x\).

Step 4 Title

Simplify the above expression to obtain the final answer. The simplification gives \(e^{-2x} - xe^{-2x} - x\) as the determinant of the given matrix.

Key Concepts

Understanding the 2x2 Matrix

A 2x2 matrix is simply a square arrangement of numbers or functions within a grid containing two rows and two columns. This specific type of matrix has four elements, which makes it relatively simple to understand compared to larger matrices. In this exercise, the matrix entries are not just numbers, but mathematical functions, which can be more complex.

When we write a 2x2 matrix, it typically looks like this:

• Top-left entry: \( a \)
• Top-right entry: \( b \)
• Bottom-left entry: \( c \)
• Bottom-right entry: \( d \)

Each of these entries can be anything from simple numbers to complex functions, as seen in the matrix from our exercise:

• \( a = e^{-x} \)
• \( b = xe^{-x} \)
• \( c = -e^{-x} \)
• \( d = (1-x)e^{-x} \)

The arrangement and relationship of these elements are essential for further calculations, such as finding the determinant, which we'll explore next.

Role of Calculus in Matrix Mathematics

Calculus is a branch of mathematics that deals with continuous change, and it plays a critical role in understanding matrices with function entries. In this context, calculus allows us to handle derivatives and integrals that might be necessary when manipulating such matrices.

In this exercise, the matrix entries involve the exponential function \(e^{-x}\), which is a common function you'll encounter in calculus, especially when working with transformations and differential equations.

• Calculus helps in finding derivatives of these entries if required by further problem conditions.
• Change of variables is often performed in calculus, which may require evaluating determinants like this one to simplify multiple integrals.

Understanding how to work with functions in the matrix form provides a solid foundation for tackling more complex mathematical problems where calculus and linear algebra intersect.

Functions as Matrix Entries

When dealing with functions as matrix entries, it is crucial to understand the properties and behaviors of these functions. In our matrix evaluation problem, each element is a function involving the exponential function \(e^{-x}\), which decreases as \(x\) increases.

Evaluating a matrix with functions involves:

• Identifying each function and its characteristics.
• Applying the rules of matrix determinants simultaneously to these functions.

A determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is found using the formula \(ad - bc\). However, since these elements are functions, you need to:

• Substitute back into the determinant formula.
• Simplify expressions while paying attention to algebraic rules applying to functions.

By applying these steps, you can unravel even complex matrices where each entry is a function, not just an ordinary number, thus enabling a deeper comprehension of mathematical transformations.