/*! 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 49 Evaluate the determinants to ver... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 49

Evaluate the determinants to verify the equation. $$\left|\begin{array}{lll} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{array}\right|=(y-x)(z-x)(z-y)$$

Short Answer

Expert verified
The determinant of the matrix verifies the equation as \((y-x)(z-x)(z-y)\).

Step by step solution

01

Understanding the matrix

The matrix contains three rows, with each column containing the variables x, y, and z raised to 0, 1, and 2 respectively.
02

Calculate the determinant

The determinant of a 3x3 matrix can be calculated using the formula: \(Det = a(ei鈭抐h)鈭抌(di鈭抐g)+c(dh鈭抏g)\). Replacing these with the elements of the given matrix, we get:\( Det = 1[(y)*z^{2}-z*y^{2}] -x [1*z^{2}-z*1] + x^{2} [1*y-y] \). With simplification this gives:\(1[z^{2}y-y^{2}z]-x[z^{2}-z]+ x^{2}[y-y]\). The term with the square falls out because it is 0. This simplifies to:\(z^{2}y-y^{2}z-z^{2}x+z^{2}+zx-yx\). Reorganizing gives us\((y-x)z^{2}+(x-z)yz+(z-y)yx\). Factor out terms to get the final answer:\((y-x)(z-x)(z-y)\).
03

Verification

The calculated determinant is exactly the same as the given equation, hence it verifies the equation.

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

Navigation A motorboat traveling with the current takes 40 minutes to travel 20 miles downstream. The return trip takes 60 minutes. Find the speed of the current and the speed of the boat relative to the current, assuming that both remain constant.

Solve the system graphically. Verify your solutions algebraically. $$\left\\{\begin{aligned} -x-y &=3 \\ x^{2}+y^{2}-4 x-21 &=0 \end{aligned}\right.$$

Operations with Matrices Find, if possible, \((a) A+B,(b) A-B,(c) 3 A,\) and \((d) 3 A-2 B.\) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{rr} 8 & -1 \\ 2 & 3 \\ -4 & 5 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 6 \\ -1 & -5 \\ 1 & 10 \end{array}\right]$$

When a system of linear equations has no solution, do the lines intersect?

You are tutoring a student in algebra. In trying to find a partial fraction decomposition, your student writes the following. $$\begin{aligned} \frac{x^{2}+1}{x(x-1)} &=\frac{A}{x}+\frac{B}{x-1} \\ x^{2}+1 &=A(x-1)+B x \\ x^{2}+1 &=(A+B) x-A \end{aligned}$$ Your student then forms the following system of linear equations. $$\left\\{\begin{aligned} A+B &=0 \\\\-A &=1 \end{aligned}\right.$$ Solve the system and check the partial fraction decomposition it yields. Has your student worked the problem correctly? If not, what went wrong?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.