/*! 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 2 True or false? \(\operatorname{d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 2

True or false? \(\operatorname{det}(A)\) is a number, not a matrix.

Short Answer

Expert verified
True, \\(\operatorname{det}(A)\\) is a number, not a matrix.

Step by step solution

01

Understanding Determinants

The determinant of a matrix is a scalar, which means it is a single number, not a matrix itself. Determinants are used with square matrices (matrices with the same number of rows and columns), and it provides information such as whether a matrix is invertible.
02

Application to the Problem

Given that the statement is about the determinant of a matrix, we must apply our understanding that the determinant, \(\operatorname{det}(A)\), represents a single numerical value. It does not alter the dimension of the original matrix; rather, it summarizes certain properties of the matrix with one number.
03

Analyzing the Statement

We need to evaluate whether the determinant \(\operatorname{det}(A)\) is a number or a matrix. Since, by definition, determinants are scalars, the statement "the determinant is a number, not a matrix" should be true.

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.

Scalar
In the context of matrices, a scalar is a single numerical value. It is distinct from vectors and matrices, which have multiple entries. When discussing the determinant of a matrix, it is important to remember that the determinant itself is a scalar. This means that no matter the complexity of the matrix, its determinant will always condense its various properties into one numerical value. For instance:
  • A scalar can be positive, negative, or zero, giving insights into the matrix it's derived from.
  • The zero value indicates a special characteristic that the matrix is not invertible, which means it does not have an inverse.
This concept is fundamental in linear algebra, and understanding it helps in grasping more complex topics involving matrices.
Square Matrices
A square matrix is a matrix with the same number of rows and columns. They are denoted by their dimensions, like 2x2, 3x3, and so on. Square matrices play a pivotal role when we talk about determinants because only square matrices have determinants.
A few key points about square matrices include:
  • The identity matrix, often denoted by I, is a good example of a square matrix. It acts as the multiplicative identity in matrix multiplication.
  • Operations like finding the trace or determinant are only possible with square matrices.
Recognizing whether a matrix is square helps determine the applicability of various matrix operations, such as calculating the determinant, which provides further insights into the matrix's properties.
Invertible Matrices
An invertible matrix, sometimes called a non-singular or non-degenerate matrix, is a matrix that has an inverse. For a matrix to be invertible, it must be square, and importantly, its determinant must not be zero.
This concept highlights:
  • If the determinant of a square matrix is zero, it suggests that the matrix is singular which means it cannot be inverted.
  • Invertible matrices are crucial in solving matrix equations, as they allow for the determination of the unique solution to a system of linear equations.
Understanding whether a matrix is invertible allows us to solve matrix equations effectively, which is an essential part of linear algebra, network analysis, and more.

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. $$\left\\{\begin{aligned} 2 x-5 y &=4 \\ x+y-z &=8 \\ 3 x &+5 z=0 \end{aligned}\right.$$

An encyclopedia saleswoman works for a company that offers three different grades of bindings for its encyclopedias: standard, deluxe, and leather. For each set that she sells, she earns a commission based on the set's binding grade. One week she sells one standard, one deluxe, and two leather sets and makes \(\$ 675\) in commission. The next week she sells two standard, one deluxe, and one leather set for a \(\$ 600\) commission. The third week she sells one standard, two deluxe, and one leather set, earning \(\$ 625\) in commission. (a) Let \(x, y,\) and \(z\) represent the commission she earns on standard, deluxe, and leather sets, respectively. Translate the given information into a system of equations in \(x, y\) and \(z\) (b) Express the system of equations you found in part (a) as a matrix equation of the form \(A X=B\). (c) Find the inverse of the coefficient matrix \(A\) and use it to solve the matrix equation in part (b). How much commission does the saleswoman earn on a set of encyclopedias in each grade of binding?

Evaluate the determinants. $$\left|\begin{array}{lllll} a & 0 & 0 & 0 & 0 \\ 0 & b & 0 & 0 & 0 \\ 0 & 0 & c & 0 & 0 \\ 0 & 0 & 0 & d & 0 \\ 0 & 0 & 0 & 0 & e \end{array}\right|$$

Evaluate the determinants. $$\left|\begin{array}{lllll} a & a & a & a & a \\ 0 & a & a & a & a \\ 0 & 0 & a & a & a \\ 0 & 0 & 0 & a & a \\ 0 & 0 & 0 & 0 & a \end{array}\right|$$

Use Cramer's Rule to solve the system. $$\left\\{\begin{aligned} 5 x-3 y+z &=6 \\ 4 y-6 z &=22 \\ 7 x+10 y &=-13 \end{aligned}\right.$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

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.