/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 Evaluate the determinant, in whi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 63

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

Short Answer

Expert verified
The determinant of the given 2x2 matrix is \(1 - \ln x\).

Step by step solution

01

Identify Matrix Elements

Identify matrix elements and organize them according to the formula for the determinant of a 2x2 matrix. In this case, \(a = x\), \(b = \ln x\), \(c = 1\), and \(d = 1/x\)
02

Apply the Formula for the Determinant of the 2x2 Matrix

Substitute the organized values back into the formula for the determinant. So, the determinant \( = (x)(1/x) - (1)(\ln x)\)
03

Simplify the Expression

By performing the multiplication and subtraction, we can simplify the determinant to \(= 1 - \ln x\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use Cramer's Rule to solve the system of linear equations, if possible. $$\begin{array}{l} 3 x_{1}+4 x_{2}=-2 \\ 5 x_{1}+3 x_{2}=4 \end{array}$$

The table below shows the numbers of subscribers \(y\) (in millions) of a cellular communications company in the United States for the years 2003 to \(2005 .\) (Source: U.S. Census Bureau)$$\begin{array}{l|c} \hline \text {Year} & \text {Subscribers} \\ \hline 2003 & 158.7 \\ 2004 & 182.1 \\ 2005 & 207.9 \\ \hline \end{array}$$ (a) Create a system of linear equations for the data to fit the curve \(y=a t^{2}+b t+c,\) where \(t\) is the year and \(t=3\) corresponds to \(2003,\) and \(y\) is the number of subscribers. (b) Use Cramer's Rule to solve your system. (c) Use a graphing utility to plot the data and graph your regression polynomial function. (d) Briefly describe how well the polynomial function fits the data.

Find (a) the characteristic equation, (b) the eigenvalues, and (c) the corresponding eigenvectors of the matrix. $$\left[\begin{array}{lll} 2 & 0 & 1 \\ 0 & 3 & 4 \\ 0 & 0 & 1 \end{array}\right]$$

Use a graphing utility or a computer software program with matrix capabilities and Cramer's Rule to solve for \(x_{1}\) if possible. $$\begin{array}{r} -0.4 x_{1}+0.8 x_{2}=1.6 \\ 2 x_{1}-4 x_{2}=5 \end{array}$$

Use a graphing utility or computer software program with matrix capabilities to find the eigenvalues of the matrix. Then find the corresponding eigenvectors. $$\left[\begin{array}{rrrr} 1 & 0 & 2 & 3 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 3 \\ 0 & -1 & 3 & 1 \end{array}\right]$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.