/*! 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 21 Evaluate the determinant of the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 21

Evaluate the determinant of the matrix. $$\left[\begin{array}{rrr} -2 & 2 & 0 \\ 0 & -1 & 1 \\ -4 & 5 & 2 \end{array}\right]$$

Short Answer

Expert verified
The determinant of the given matrix is -14.

Step by step solution

01

Find the products of the diagonals

We will multiply the elements from top left to bottom right diagonally and add them. So, it's (-2*-1*2) + (2*1* -4) + (0*0*5) = 4 - 8 + 0 = -4
02

Find the products of the reversed diagonals

We will now multiply the elements from top right to bottom left diagonally and add them. So, it's (0*-1* -4) + (2*1*5) + (-2*0*2) = 0 + 10 + 0 = 10
03

Subtract the products of reversed diagonals from products of diagonals

The last step is to calculate the determinant by taking the difference of the results from step 1 and 2. So, determinant = -4 - 10 = -14

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!

Key Concepts

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

Diagonal products in determinants
In linear algebra, finding the determinant of a matrix is an important operation. For a 3x3 matrix, one method to calculate this is by using diagonal products. This involves multiplying the elements along the diagonals of the matrix and summing them up.
  • First, multiply the elements of the main diagonal, which stretches from the top-left corner to the bottom-right corner. This is often referred to as the product of the forward diagonal.
  • Then, consider the "reversed" diagonal terms, stretching from the top-right to the bottom-left corner. This corresponds to calculating the products along these diagonals as well.
By finding both sets of diagonal products, the determinant can be obtained by subtracting the sum of the reversed diagonal products from the sum of the forward diagonal products.
Understanding a 3x3 matrix
A 3x3 matrix is a key concept in linear algebra, consisting of three rows and three columns. This setup is important for various calculations, including solving systems of equations and understanding transformations. In the given problem, the matrix is:\[\begin{bmatrix}-2 & 2 & 0 \0 & -1 & 1 \-4 & 5 & 2 \end{bmatrix}\]Each element in the matrix belongs to a specific row-column intersection. In problems involving determinants, recognizing how these elements interact along the diagonals helps in performing systematic calculations, like those needed in the method described above.
Linear algebra and its applications
Linear algebra is the branch of mathematics concerning linear equations, matrices, and vector spaces. It provides the tools for understanding broad mathematical concepts and for solving practical problems involving multi-dimensional spaces. Understanding determinants, like those from a 3x3 matrix, is foundational for:
  • Solving systems of simultaneous equations.
  • Performing important calculations in computer graphics, such as transformations and rotations.
  • Determining characteristics properties of linear maps represented by matrices.
By mastering these concepts, students gain powerful techniques applicable in numerous scientific, engineering, and economic fields.

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

Solve the optimization problem. Minimize \(P=20 x+30 y\) subject to the following constraints. $$ \left\\{\begin{aligned} 3 x+y & \leq 9 \\ y & \geq x \\ y & \geq 2 \\ x & \geq 0 \end{aligned}\right. $$

Consider the following augmented matrix. For what value(s) of \(a\) does the corresponding system of linear equations have infinitely many solutions? One solution? Explain your answers.$$\left[\begin{array}{lll|r}1 & 0 & 0 & -2 \\\0 & 1 & 0 & 5 \\\0 & 0 & a & 0\end{array}\right]$$.

Sarah can't afford to spend more than \(\$ 90\) per month on transportation to and from work. The bus fare is only \(\$ 1.50\) one way, but it takes Sarah 1 hour and 15 minutes to get to work by bus. If she drives the 15 -mile round trip, her one-way commuting time is reduced to 40 minutes, but it costs her S.40 per mile. If she works at least 20 days a month, how often does she have to drive in order to minimize her commuting time and keep within her monthly budget?

$$\begin{aligned}&\text { If } A=\left[\begin{array}{cc}4 a+5 & -1 \\\\-4 & -7\end{array}\right] \text { and } B=\left[\begin{array}{rr}7 & 0 \\\\-4 & -8\end{array}\right], \text { for what }\\\&\text { value(s) of } a \text { does } 2 B-3 A=\left[\begin{array}{ll}2 & 3 \\\4 & 5\end{array}\right] ?\end{aligned}$$

Find \(A^{2}\) (the product \(A A\) ) and \(A^{3}\) (the prod\(\left.u c t\left(A^{2}\right) A\right)\). $$A=\left[\begin{array}{rr}2 & -1 \\\1 & 0\end{array}\right]$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

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.