Find Determinant B

harpazo

harpazo

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Use the properties of determinants to find the value of determinant B if it is known that determinant A = 4.

NOTE: The textbook does not show how this is done. Can someone help me by providing a step by step solution?

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The book has almost certainly taught you how row operations affect a determinant. You need to use those ideas (which are the properties referred to in the problem).

What happens to the determinant if you swap two rows?

What happens to the determinant if you double a row?
 
harpazo said:
Use the properties of determinants to find the … determinant [of matrix] B … The textbook does not show how …
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Let's start at the beginning. Does Sullivan's text show the properties of determinants? If so, what are they?

?
 
You said "Using the properties of determinants". What properties of determinants do you know? Isn't there one that says what happens if one row is multiplied by two?
 
harpazo said:
Use the properties of determinants to find the value of determinant B if it is known that determinant A = 4.
NOTE: The textbook does not show how this is done. Can someone help me by providing a step by step solution?
Click to expand...
I happen to know Sullivan's tricks. This is topical.
The determinate \(\displaystyle |A|=1\cdot\left| {\begin{array}{*{20}{c}} y&z \\ v&w \end{array}} \right| - 2\left| {\begin{array}{*{20}{c}} x&z \\ u&w \end{array}} \right| + 3\left| {\begin{array}{*{20}{c}} x&y \\ u&v \end{array}} \right|\) is achieved by expanding along the third row.
Now do the same for ∣B∣\displaystyle |B| and compare.
 
I just checked in my copy of Sullivan's Precalculus (section 10.3) for comparison, and found this exact problem. This section has an objective (the last) headed "Know Properties of Determinants", which gives five theorems, two of which are the ones needed for this problem:
  • The value of a determinant changes sign if any two rows (or any two columns) are interchanged.
  • If any row (or any column) of a determinant is multiplied by a nonzero number k, the value of the determinant is also changed by a factor of k.
Nothing more is needed. There is no example exactly like this one, but the theorems are demonstrated using specific numbers. Students are expected to be able to apply the theorems without additional help.
 
pka said:
I happen to know Sullivan's tricks. This is topical.
The determinate \(\displaystyle |A|=1\cdot\left| {\begin{array}{*{20}{c}} y&z \\ v&w \end{array}} \right| - 2\left| {\begin{array}{*{20}{c}} x&z \\ u&w \end{array}} \right| + 3\left| {\begin{array}{*{20}{c}} x&y \\ u&v \end{array}} \right|\) is achieved by expanding along the third row.
Now do the same for ∣B∣\displaystyle |B| and compare.
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This is a good start.
 
HallsofIvy said:
You said "Using the properties of determinants". What properties of determinants do you know? Isn't there one that says what happens if one row is multiplied by two?
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I did read the text about properties of determinants. However, the wording used by Sullivan, like most math and science books, is not too clear. Sullivan is super clear to you and other mathematicians here, but to someone like me who is reviewing material learned over 20 years ago, the jargon used is a bit fuzzy.
 
Yes, the problem with any textbook is that the author doesn't know you, and has to assume you are learning at a certain rate so that after a while things can be said more briefly. Rarely can they afford the space to give all the details one would like.

Of course, teachers, tutors, or sites like this are a way to fill the gap between what the author can fit, and what an individual needs. Ideally, you would state what you understand, and ask for help at particular points. In this case, that might be to recognize that the question asks you to apply properties, to find those properties and state them (as in some cases authors might vary in the properties they list), and ask how those properties apply to this problem, or what they mean. Then tutors can help you understand those details. Three of us, in effect, asked you to do that.

Note, by the way, that pka's way doesn't use the properties, and is the long way around, if you gave up on the properties.

In any case, do you understand them yet? That's the important thing, as that is the purpose of the problem.
 
Dr.Peterson said:
… the problem with any textbook is that the author doesn't know you … Ideally, you would state what you understand, and ask for help at particular points … Then tutors can help you understand those details. Three of us, in effect, asked you to do that … In any case, do you understand them yet? That's the important thing …
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I like Michael Sullivan. David Cohen is very good as well. I decided to purchase several EASY TO READ math books by Michael Kelly. Kelly greatly reduces math jargon. Kelly's books are like the DUMMIES series.
 
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what they are after is

B=(100010002)A\displaystyle B = \begin{pmatrix}1&0&0\\0&1&0\\0&0&2\end{pmatrix}A

∣B∣=∣(100010002)∣∣A∣=2⋅4=8\displaystyle |B| = \left|\begin{pmatrix}1&0&0\\0&1&0\\0&0&2\end{pmatrix}\right||A| = 2\cdot 4 = 8

This is essentially what Dr. Peterson was getting at in post #2
 
A "property of determinants": if every number in a single row (or column) of a matrix is multiplied by the same number then the value of the determinant is multiplied by that number. Here "B" is the same as "A" except that the third row is multiplied by 2. Since the determinant of A is 4, the determinant of B is 2(4)= 8.
 
HallsofIvy said:
A "property of determinants": if every number in a single row (or column) of a matrix is multiplied by the same number then the value of the determinant is multiplied by that number. Here "B" is the same as "A" except that the third row is multiplied by 2. Since the determinant of A is 4, the determinant of B is 2(4)= 8.
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I got it. Thanks.
 
Otis said:
Why are you posting linear algebra exercises and asking us to teach you the steps?

:confused:
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1. Sullivan has a chapter on the basics of matrices and determinants in his college algebra text. Did you read what I just said? I said college algebra text not a linear algebra text. I am not crazy enough to jump into advanced algebra for no reason.

2. I am not going to skip chapters in the Sullivan textbook.

3. This material is not too bad. I simply asked for guidance, tips, etc.

4. You are the only person that loves to deeply look for reasons to start a problem here. Then, when I defend myself, you know, give reasons for posting questions, others take your side against me and the entire cycle begins.

Tip:

WHEN YOU SEE HARPAZO'S QUESTIONS, ANNOYING OR NOT, SKIP THE POST. I just made your life here so much easier. This is what I would do with "annoying" people like HARPAZO.
 
harpazo said:
1. … Did you read what I just said? …
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Yes.


I simply asked for guidance, tips, etc.
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No, you did not. You asked for a "step-by-step solution".


… WHEN YOU SEE HARPAZO'S QUESTIONS, ANNOYING OR NOT, SKIP THE POST …
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Generally, it's not a matter of annoyance. It's a matter of the forum's guidelines and why you're still not following them. It seems like you still want to create your questions archive, for the benefit of other students. You're certainly capable of answering a lot of your own questions, without having to post them (especially all the repeats).

You have been banned at other sites for disruption. I ask questions to help me better understand what you're trying to do here.

  \;
 
Otis said:
Yes.



No, you did not. You asked for a "step-by-step solution".



Generally, it's not a matter of annoyance. It's a matter of the forum's guidelines and why you're still not following them. It seems like you still want to create your questions archive, for the benefit of other students. You're certainly capable of answering a lot of your own questions, without having to post them (especially all the repeats).

You have been banned at other sites for disruption. I ask questions to help me better understand what you're trying to do here.

  \;
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1. I get most of the chapter questions right working on my own using paper and a calculator.

2. I only post problems that

A. I have not been able to solve on my own
B. I find interestingly challenging
C. I do not even know how to begin, for example, proofs.

3. When you see my questions ask yourself the followings:

A. How many questions in the current textbook section did harpazo answer correctly?
B. How many questions per section does harpazo actually and rigorously answers on paper?
C. Should I give him credit for trying to review material he learned back in the 1980s and early 1990s as a CUNY student?
D. Is harpazo looking for trouble with members here?
E. Is harpazo flooding our site with stupid math problems that he is not trying, really trying hard to do on his own?

TRUST ME: I TRY, I REALLY TRY hard to learn the material well. You see, I enjoy mathematics but it's not the only hobby I enjoy.
 
harpazo said:
… When you see my questions ask yourself …
How many questions in the current textbook section did harpazo answer correctly? …
How many questions per section does harpazo actually and rigorously answers on paper? …
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I don't understand why you've made this request. You're the only person who knows what you've done outside the forum. Why should tutors ask themselves about stuff that we have no way to see or confirm?

You could ask yourself a question, before submitting your posts: Did I follow the forum's guidelines.

Whenever you cannot remember what the guidelines are, then I ask you to review them and to make any necessary additions or edits before submitting your posts.

  \;
 
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