Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

In Exercises 27 and $28, A$ and $B$ are $n \times n$ matrices. Mark each statement True or False. Justify each answer.a. A row replacement operation does not affect the determinant of a matrix.b. The determinant of $A$ is the product of the pivots in any echelon form $U$ of $A,$ multiplied by $(-1)^{r},$ where $r$ is the number of row interchanges made during row reduction from $A$ to $U .$c. If the columns of $A$ are linearly dependent, then det $A=0$ .d. $\operatorname{det}(A+B)=\operatorname{det} A+\operatorname{det} B$

a. Trueb. Truec. Trued. False.

Algebra

Chapter 3

Determinants

Section 2

Properties of Determinants

Introduction to Matrices

Missouri State University

Harvey Mudd College

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

02:38

In Exercises 27 and $28, A…

00:23

In Exercises 21 and $22, A…

12:10

02:01

In Exercises 39 and $40, A…

09:01

In Exercises 21 and $22,$ …

04:42

In Exercises 17 and 18 , $…

06:37

03:53

In Exercises 25 and $26, A…

07:24

In Exercises 25 and $26,$ …

02:07

In Exercises 21 and 22, ma…

were given statements about to end by an matrices A and B and whereas to march mark these statements as true or false and justify our answer. So the statement in part A is that a row replacement operation does not affect the determinant of the matrix. This is true to see why refer to fear um, three of the book. This is one of the properties listed in part beef. This statement is the determinant of a is the product of the pivots. In any echelon form you of a multiplied by negative ones. They are where r is the number of row interchanges made during row reduction from a to the echelon form you. This is also true. And to understand why the book actually explains this pretty well in the paragraph after example two of this section to summarize what that paragraph says though, when we're putting the matrix a into an echelon form you, we're really only performing row replacement operations and making row interchanges. And so the only effects would be their own to changes on the determinant which for each wrong to change, we'll have to multiply by negative one. So if there's are running to changes that results in negative ones. They are times the determinant. Then in part c, our statement is if the columns of a are linearly dependent in the determinant of a is zero. This is also true. Once again, this was actually explained in the paragraph this time following the're, um four. But to summarize, we know that if the calls of Ireland nearly dependent mhm, this would mean that one of them is a scaler multiple of all the other problems. And so if that's true, then using the permanent properties we can reduce that column to zero and then we have a zero column and our determinant, which means the determinant would be zero and finally in part d. The statement is the determinant. A plus B is equal to the determinant of a plus the determinant to be so claiming that the determinant is additive. This is false. Actually, Now we do know the determinant is multiplication. However, there was actually an example following example five where they give an example of two matrices such that there's some. The determinant of their sum is not equal to the sum of their determinants. So if you want. Think of your own example here, or I could give you one. Consider the matrices the identity matrix and also the matrix. 0110 So we'll take a equal to 1001 and be equal to 0110 Actually, it doesn't work. What on? Actually. Instead, let's take a to be the matrix 1101 and we'll take B two b the matrix 10 11 Then we have that the determinants of a plus. The determinant to be is equal to Well, this is going to be one plus one or two. On the other hand, the determinant of a plus B Well, this is the determinant of the matrix 21 12 or four minus one, which is three. So we have here an example where the determinant of a plus the determinant of B is not equal to determined to be a plus B

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, the absolute value or modulus |x| of a real number x is its …

In Exercises 27 and $28, A$ and $B$ are $n \times n$ matrices. Mark each sta…

In Exercises 21 and $22, A$ and $B$ are $n \times n$ matrices. Mark each sta…

In Exercises 39 and $40, A$ is an $n \times n$ matrix. Mark each statement T…

In Exercises 21 and $22,$ mark each statement True or False. Justify each an…

In Exercises 17 and 18 , $A$ is an $m \times n$ matrix. Mark each statement …

In Exercises 25 and $26, A$ denotes an $m \times n$ matrix. Mark each statem…

In Exercises 25 and $26,$ mark each statement True or False. Justify each an…

In Exercises 21 and 22, mark each statement True or False. Justify each answ…

06:07

In Exercises 7–10, determine the values of the parameter s for which the sys…

02:24

Let $H$ be the set of points inside and on the unit circle in the $x y$ -pla…

01:52

In Exercises $33-36,$ verify that det $E A=(\operatorname{det} E)(\operatorn…

02:04

In Exercises $27-30$ , use coordinate vectors to test the linear independenc…

01:40

In Exercises 31–36, mention an appropriate theorem in your explanation.L…

02:58

[M] Repeat Exercise 43 with the matrices $A$ and $B$ from Exercise 42 . Then…

02:53

Is $\left[\begin{array}{r}{4} \\ {-3} \\ {1}\end{array}\right]$ an eigenvect…

02:03

Verify that det $A B=(\operatorname{det} A)(\operatorname{det} B)$ for the m…

01:48

Determine if the given set is a subspace of $\mathbb{P}_{n}$ for an appropri…

02:15

Determine if $\mathbf{w}=\left[\begin{array}{r}{1} \\ {3} \\ {-4}\end{array}…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.