/*! 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 1 Both det \((A)\) and \(|A|\) rep... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 1

Both det \((A)\) and \(|A|\) represent the __________ of the matrix \(A\).

Short Answer

Expert verified
The terms det(A) and |A| both represent the determinant of the matrix A.

Step by step solution

01

Understand the concept

Consider the terms det(A) and |A|. In a matrix, both represent the same thing: they are used to signify the determinant of a matrix.
02

Identify the determinant of a matrix

The determinant is a special number that is calculated from a square matrix. It has a wide array of applications, and is used while finding the solution of a system of linear equations (Cramer's Rule), in calculus, etc.

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.

Matrix Algebra
Matrix algebra is a branch of mathematics that extends the concepts of addition, subtraction, multiplication, and division to matrices. In simpler terms, it's like regular algebra, but with matrices instead of numbers. Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns.
  • Matrix algebra includes operations such as matrix addition, subtraction, and multiplication.
  • For multiplication, the key is that elements are matched in a specific way, leading to the next matrix's element construction.
  • Algebraic processes are foundational for vector spaces, which are crucial in many areas such as physics and engineering.
An important aspect of matrix algebra is finding the determinant of a matrix, which is essential in many matrix computations. The determinant is a special number calculated from a square matrix.
Cramer's Rule
Cramer's Rule is a theorem used to solve linear equations using determinants. It is an elegant and straightforward method for finding the solution of a system of linear equations with as many equations as unknowns.
  • This method involves calculating the determinant of the coefficient matrix and then using it to find the determinants of matrices where one column has been replaced by the constants from the equation.
  • The solution for each variable in the system of equations is then found as the ratio of these determinants.
  • It is important to note that Cramer's Rule is applicable only when the determinant of the coefficient matrix is non-zero, ensuring a unique solution.
Using Cramer's Rule can be quite efficient with small matrices but becomes less practical with larger ones due to the complexity of determinant calculations.
Linear Equations
Linear equations are mathematical expressions that represent a line when graphed on a coordinate plane. These equations can have one or more variables, but each term is either a constant or a product of a constant and a single variable.
  • The general form of a linear equation in two variables is: \( ax + by = c \).
  • In matrix form, a system of linear equations can be expressed as \( A \mathbf{x} = \mathbf{b} \), where \( A \) is the matrix of coefficients, \( \mathbf{x} \) is the column of variables, and \( \mathbf{b} \) is the column of constants.
  • Solutions to linear equations can be found using various methods like substitution, elimination, and matrix methods such as using determinants.
The determinant is an essential tool here as it helps in methods like Cramer's Rule to find the solutions of linear equations efficiently, especially when expressed in a matrix format.

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

Writing a Matrix in Row-Echelon Form, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.) $$\left[ \begin{array}{rrrr}{1} & {-3} & {0} & {-7} \\ {-3} & {10} & {1} & {23} \\ {4} & {-10} & {2} & {-24}\end{array}\right]$$

Solving an Equation In Exercises \(81-88,\) solve for \(x .\) $$\left| \begin{array}{cc}{x+4} & {-2} \\ {7} & {x-5}\end{array}\right|=0$$

Curve Fitting, use a system of equations to find the quadratic function \(f(x)=a x^{2}+b x+c\) that satisfies the given conditions. Solve the system using matrices. $$f(1)=9, f(2)=8, f(3)=5$$

Properties of Determinants In Exercises \(101-103\) ,a property of determinants is given \((A\) and \(B\) are squarematrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. Properties of Determinants In Exercises \(101-103\) ,a property of determinants is given \((A\) and \(B\) are squarematrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. If \(B\) is obtained from \(A\) by interchanging two rows of \(A\) or interchanging two columns of \(A,\) then \(|B|=-|A|\) $$(a)\left| \begin{array}{rrr}{1} & {3} & {4} \\ {-7} & {2} & {-5} \\ {6} & {1} & {2}\end{array}\right|=-\left| \begin{array}{rrr}{1} & {4} & {3} \\ {-7} & {-5} & {2} \\ {6} & {2} & {1}\end{array}\right|$$ $$(b)\left| \begin{array}{r|rrr}{1} & {3} & {4} \\ {-2} & {2} & {0} \\ {1} & {6} & {2}\end{array}\right|=-\left| \begin{array}{rrr}{1} & {6} & {2} \\ {-2} & {2} & {0} \\ {1} & {3} & {4}\end{array}\right|$$

Interpreting Reduced Row-Echelon Form , an augmented matrix that represents a system of linear equations (in variables \(x, y,\) and \(z,\) if applicable) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$\left[ \begin{array}{rrrr}{1} & {0} & {0} & {\vdots} & {-4} \\ {0} & {1} & {0} & {\vdots} & {-10} \\ {0} & {0} & {1} & {\vdots} & {4}\end{array}\right]$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.