Matrix Homework: Solving f(A)=0 with A^5 | Help with Polynomial Determinant

[gtfitzpatrick]

Homework Statement

if A=

[
1 2 1
2 3 1
3 5 3 ]

and f(x) is the polynomial given by
f(x) =
det
1-x 2 1
2 3-x 1
3 5 3-x

show that f(A) = 0. Evaluate A^5do i get the determinant of A and put into the det of f(x)?
Not really sure how to go about this?

 

[D H]

The way to go about this is to determine the characteristic polynomial f(x) and then evaluate f(A). There is a bit of a trick in doing this. The characteristic polynomial is obviously a cubic: f(x)=a_0+a_1x+a_2x^2+a_3x^3. Write this as f(x)=a_0x^0+a_1x^1+a_2x^2+a_3x^3.

The trick: Just as any non-zero scalar taken to the zeroth power is 1, a non-singular square matrix raised to the zeroth power is the identity matrix.

 

[kof9595995]

Just for further reading, you can check out Cayley-Hamilton's theorem on wiki. It's relevant but not necessary to solve the problem.

 

[gtfitzpatrick]

im not following what u are doing but i got the det of f(x) which i got to be -x^3+x^2-3x-1
so I am figuring what i ab going to do now is sub in A and do all the sums and hope it = 0, is this wrong?

 

[gtfitzpatrick]

ok my approch didnt work, could u explain the trick bit to be again please?

 

[paxprobellum]

im not following what u are doing but i got the det of f(x) which i got to be -x^3+x^2-3x-1
so I am figuring what i ab going to do now is sub in A and do all the sums and hope it = 0, is this wrong?

It looks like you dropped a 7 in your determinant calculation. I got:

f(x) = -x^3 + 7*x^2 - 3*x -1

so f(A) = ?

 

[Дьявол]

ok my approch didnt work, could u explain the trick bit to be again please?

For the first problem solve both determinants and substitute the solution of the det. A for "x" in the second determinant.

For A⁵, use the value that you have found before and just put it on power 5.

Regards.

 

[gtfitzpatrick]

ahhhhhhh yes thanks a mill that got it,
evaluate A^5 there is obviously a trick to this, surely i can't have to work it all out?

 

[gtfitzpatrick]

oh right see the other reply now, thanks a mill

 

[gtfitzpatrick]

right still on this, i have the det of f(x) = -x^3 + 7*x^2 - 3*x -1 and i subed in matrix A, and did the sums and it = 0, which is what i should get. then i tried what Дьявол said and got the det of A which i got to be -1 and subed this value back into det of f(x) and i got an answer of 10, which isn't right, and then which value do i put ^5?

 

[Дьявол]

right still on this, i have the det of f(x) = -x^3 + 7*x^2 - 3*x -1 and i subed in matrix A, and did the sums and it = 0, which is what i should get. then i tried what Дьявол said and got the det of A which i got to be -1 and subed this value back into det of f(x) and i got an answer of 10, which isn't right, and then which value do i put ^5?

det(A)=\begin{vmatrix}<br /> 1 &amp; 2 &amp; 1\\ <br /> 2 &amp; 3 &amp; 1\\ <br /> 3 &amp; 5 &amp; 3<br /> \end{vmatrix}=-1<br />

So det(A)⁵=(-1)⁵, right?

Regards.

 

[VietDao29]

A=\begin{vmatrix}<br /> 1 &amp; 2 &amp; 1\\ <br /> 2 &amp; 3 &amp; 1\\ <br /> 3 &amp; 5 &amp; 3<br /> \end{vmatrix}=-1<br />

So A⁵=(-1)⁵, right?

Regards.

Дьявол, I don't mean to be rude, or anything, but do you really know what you are saying?? Did you actually solve the problem? Can you really solve it?

Unless you want to confuse the OP with what you "think" to be true, please don't post anything. Please only help people, if you actually know how to help!

Everyone needs to learn, this is true for all of us. If that's a new problem for you, and you are not even sure how to solve it. Please stand by and listen. You cannot pretend to be know-it-all, while in fact, you don't know anything!.. :|

I know, guiding other people, and helping them is very nice. Just make sure that you post don't make other people (especially the OP, who's needing help the most) confused.

right still on this, i have the det of f(x) = -x^3 + 7*x^2 - 3*x -1 and i subed in matrix A, and did the sums and it = 0, which is what i should get. then i tried what Дьявол said and got the det of A which i got to be -1 and subed this value back into det of f(x) and i got an answer of 10, which isnt right, and then which value do i put ^5?

He did solve one part of the problem, but did you??

-----------------------

@OP: A is not a determinant, it's a matrix, you just need to sub A in the expression:

f(x) = -x³ + 7x² - 3x -1

And you'll have f(A) = 0.

To calculate A⁵, you can do as http://en.wikipedia.org/wiki/Cayley–Hamilton_theorem#Examples".

Read from:

Examples

[...]

The Cayley–Hamilton theorem then claims that
A^2 - 5A - 2I_2 = \left( \begin{array}{ll} 0 &amp; 0 \\ 0 &amp; 0 \end{array} \right)

Then, for example, to calculate A⁴, observe

[...]

 

[Дьявол]

Дьявол, I don't mean to be rude, or anything, but do you really know what you are saying?? Did you actually solve the problem? Can you really solve it?

Unless you want to confuse the OP with what you "think" to be true, please don't post anything. Please only help people, if you actually know how to help!

Everyone needs to learn, this is true for all of us. If that's a new problem for you, and you are not even sure how to solve it. Please stand by and listen. You cannot pretend to be know-it-all, while in fact, you don't know anything!.. :|

I know, guiding other people, and helping them is very nice. Just make sure that you post don't make other people (especially the OP, who's needing help the most) confused.
He did solve one part of the problem, but did you??

-----------------------

@OP: A is not a determinant, it's a matrix, you just need to sub A in the expression:

f(x) = -x³ + 7x² - 3x -1

And you'll have f(A) = 0.

To calculate A⁵, you can do as http://en.wikipedia.org/wiki/Cayley–Hamilton_theorem#Examples".

Read from:

@VietDao29

Could you please possibly explain what was wrong with my statement?

Maybe the OP is not so experienced, to multiply matrices.

I guess he need to solve the determinant and power it to 5.

He got 2 separate problems, and he mixed them up. That's why he is confused, not because of my statement.

Why did you say that I lack of knowledge? What did I do wrong? What is your level of knowledge?

Regards.

Edit: Aahh, I see, you're are telling me about det(A)⁵ and not A⁵. I just missed the notation det. Sorry about it, it was typo.

Now, do you think that he would understand anything from the Cayley-Hamilton theorem?

I would better find A^2=A \cdot A and then A^5=A^2 \cdot A^2 \cdot A

Look, I always try to make the things as simple as possible. ;)

 

[HallsofIvy]

There was an initial misunderstanding that you may have compounded.

He gave a matrix A, a function f(x) and asked about showing that f(A)= 0. Also then asked about calculating A⁵.

His misunderstanding was when he said "do i get the determinant of A and put into the det of f(x)?" There is nothing in the problem that says anything about the determinants of A and you may have compounded the misunderstanding by saying, yes, he should find the determinant of A and put that into f(x) (which is specifically given by a determinant). It is A itself that should be put into f(x). But the real point is that every matrix satisifes its own characteristic equation. Since f(x)= 0 is the characteristic equation of A, if he has that theorem, the problem is easy. That's one difficulty not showing any work- we have no idea if he has already seen that theorem or if this exercise is asking him to actually calculate f(x) and then calculate f(A) as a preliminary to that theorem.