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Chapter 8: Problem 4

Compute the determinant of the given matrix. (Some of these matrices appeared in Exercises \(1-8\) in Section 8.4.) \(L=\left[\begin{array}{cc}\frac{1}{x^{3}} & \frac{\ln (x)}{x^{3}} \\\ -\frac{3}{x^{4}} & \frac{1-3 \ln (x)}{x^{4}}\end{array}\right]\)

Short Answer

Expert verified
The determinant of the matrix is \( \frac{1}{x^7} \).

Step by step solution

01

Understand the Formula for Determinant

The formula to compute the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is \( ad - bc \).
02

Identify Matrix Elements

In matrix \( L = \begin{pmatrix} \frac{1}{x^3} & \frac{\ln(x)}{x^3} \ -\frac{3}{x^4} & \frac{1-3\ln(x)}{x^4} \end{pmatrix} \), we have: \( a = \frac{1}{x^3}, b = \frac{\ln(x)}{x^3} , c = -\frac{3}{x^4}, d = \frac{1-3\ln(x)}{x^4} \).
03

Calculate the Products

Compute \( ad \) and \( bc \):For \( ad \):\[ ad = \left( \frac{1}{x^3} \right) \left( \frac{1-3\ln(x)}{x^4} \right) = \frac{1 - 3\ln(x)}{x^7} \]For \( bc \):\[ bc = \left( \frac{\ln(x)}{x^3} \right) \left( -\frac{3}{x^4} \right) = -\frac{3\ln(x)}{x^7} \]
04

Substitute and Simplify the Determinant Expression

Use the determinant formula \( ad - bc \) to find:\[ ad - bc = \frac{1 - 3\ln(x)}{x^7} - \left(-\frac{3\ln(x)}{x^7}\right) \]This simplifies to:\[ \frac{1 - 3\ln(x) + 3\ln(x)}{x^7} = \frac{1}{x^7} \]
05

Conclude with the Determinant

The determinant of the matrix \( L \) is calculated as \( \frac{1}{x^7} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrices
Matrices are rectangular arrays of numbers arranged in rows and columns. Learning to work with matrices is an important skill in linear algebra. The size of a matrix is described by listing the number of rows followed by the number of columns it contains, such as 2x2, 3x3, etc.
Matrices are used extensively in various fields, such as computer graphics, physics, and economics, to represent and solve systems of linear equations.
  • In a 2x2 matrix, which has 2 rows and 2 columns, elements are usually represented as: \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \).
  • Each position in the matrix holds a specific number, allowing us to organize data or coefficients systematically.
  • The arrangement helps in performing operations like addition, subtraction, multiplication, and finding determinants.
Determinant Calculation
The determinant is a scalar value that can be computed from the elements of a square matrix. It provides valuable information about the matrix, such as whether it is invertible or not. For a 2x2 matrix, the determinant is calculated using a simple formula: \( ad - bc \).
  • The determinant helps in understanding properties of the matrix, like whether a matrix has an inverse.
  • Determinant calculations are crucial when solving linear systems using methods like Cramer's rule.
  • Finding the determinant involves identifying each element correctly and applying the formula accurately.
Matrix Algebra
Matrix algebra involves operations such as addition, subtraction, and multiplication among matrices. It is a fundamental aspect of linear algebra. These operations follow specific rules that differ from regular arithmetic operations.
Understanding how these algebraic operations work will help you manipulate matrices and solve related problems effectively.
  • Addition/subtraction of matrices: Only possible when matrices are of the same size. Perform it element by element.
  • Matrix multiplication: More complex; rows of the first matrix are multiplied with the columns of the second, summing the products.
  • Matrices can also be inverted and transposed, which are important transformations in many applications.
2x2 Matrix Formula
The determinant formula for a 2x2 matrix is one of the simplest yet powerful tools in matrix algebra. This formula, \( ad - bc \), provides a shortcut to determine particular characteristics of a matrix quickly.
Let's look at the steps in applying this formula for determinants with an example:
  • Consider a 2x2 matrix: \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \).
  • Identify elements: \( a, b, c, d \) as per their positions.
  • Apply the determinant formula: Calculate \( ad \) and \( bc \). Subtract to get the determinant: \( ad - bc \).
  • This calculation tells us if the matrix is singular (determinant is zero) or non-singular (determinant is non-zero), which affects its invertibility.

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Most popular questions from this chapter

How much of a 5 gallon \(40 \%\) salt solution should be replaced with pure water to obtain 5 gallons of a \(15 \%\) solution?

Solve the given system of nonlinear equations. Use a graph to help you avoid any potential extraneous solutions. $$ \left\\{\begin{aligned} x^{2}+y^{2} &=25 \\ y-x &=1 \end{aligned}\right. $$

A local buffet charges \(\$ 7.50\) per person for the basie buffet and \(\$ 9.25\) for the deluxe buffet (which includes crab legs.) If 27 diners went out to eat and the total bill was \(\$ 227.00\) before taxes, how many chose the basic buffet and how many chose the deluxe buffet?

In Exercises \(23-29,\) consider the following scenario. In the small village of Pedimaxus in the country of Sasquatchia, all 150 residents get one of the two local newspapers. Market research has shown that in any given week, \(90 \%\) of those who subscribe to the Pedimaxus Tribune want to keep getting it, but \(10 \%\) want to switch to the Sasquatchia Picayune. Of those who receive the Picayune, \(80 \%\) want to continue with it and \(20 \%\) want switch to the Tribune. We can express this situation using matrices. Specifically, let \(X\) be the 'state matrix' given by $$ X=\left[\begin{array}{l} T \\ P \end{array}\right] $$ where \(T\) is the number of people who get the Tribune and \(P\) is the number of people who get the Picayune in a given week. Let \(Q\) be the 'transition matrix' given by $$ Q=\left[\begin{array}{ll} 0.90 & 0.20 \\ 0.10 & 0.80 \end{array}\right] $$ such that \(Q X\) will be the state matrix for the next week. Let's assume that when Pedimaxus was founded, all 150 residents got the Tribune. (Let's call this Week \(0 .)\) This would mean $$ X=\left[\begin{array}{r} 150 \\ 0 \end{array}\right] $$ Since \(10 \%\) of that 150 want to switch to the Picayune, we should have that for Week 1, 135 people get the Tribune and 15 people get the Picayune. Show that \(Q X\) in this situation is indeed $$ Q X=\left[\begin{array}{r} 135 \\ 15 \end{array}\right] $$

Carl's Sasquatch Attack! Game Card Collection is a mixture of common and rare cards. Each common card is worth $$\$ 0.25$$ while each rare card is worth $$\$ 0.75 .$$ If his entire 117 card collection is worth $$\$ 48.75,$$ how many of each kind of card does he own?
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