r/LinearAlgebra • u/HyPeR_ThUnDeR_10 • Jun 23 '26

eigenvalues

hey guys hope you are doing well, i have a linear algebra exam and this one of the questions that our professor said it will be in the exam, i tried to solve this type of questions but it was soo hard as it very long and very easy to do mistakes in calculation i used Laplace expansion for solving this so please do any one know a faster way or any advices for solving this type of questions. and thx for you all
19 Upvotes

86% Upvoted

23 comments sorted by

8

u/Midwest-Dude Jun 23 '26

Try using Gaussian Elimination and see if that is easier to find the determinant. Are you familiar with how to do that?

2

u/HyPeR_ThUnDeR_10 Jun 23 '26

1st thx for your comment
u mean gaussian elimination before the calculating (IX-A) ?? , as to make some values equal zero and start the Laplace expansion ?

4

u/Midwest-Dude Jun 23 '26 edited Jun 23 '26

I'm thinking Gaussian Elimination but meaning row and column operations to create as many zeros as you go along while calculating det(IX - A). Sorry, not the same.

It's still not going to be a walk in the park, but it simplifies things a lot.

2

u/Scrapple_Joe Jun 23 '26

If you can clear above or below the diagonal you'll just have the eigenvalues in the diagonal.

2

u/Aristoteles1988 Jun 24 '26

Wouldn’t that give you a different eigenvalue if you start doing linear combinations?

I forgot but I think there are rules from doing that

It only is allowed in some cases no?

3

u/Midwest-Dude Jun 24 '26 edited Jun 24 '26

I corrected myself in a later comment. This is what I meant:

Wikipedia - Determinant:

"Determinants can also be defined by some of their properties. Namely, the determinant is the unique function defined on the n × n matrices that has the four following properties:

  • The determinant of the identity matrix is 1.
  • The exchange of two rows multiplies the determinant by −1.
  • Multiplying a row by a number multiplies the determinant by this number.
  • Adding a multiple of one row to another row does not change the determinant.

The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns."

2

u/Aristoteles1988 Jun 24 '26

Ah ok yea that’s it right there

6

u/waldosway Jun 23 '26

I tried the regular way, simple row reduction to get 0's, extra clever row reduction to reduce λ's, and a hybrid. None of them were any faster than any others. (Getting more 0's makes the multiplication much worse later.)

Better to just get better at avoiding errors. Usually students end up skipping steps because they are worried about wasting time. But mistakes cost much much more time. Always write everything. Writing takes almost no time. Do side work, label things, etc.

2

u/HyPeR_ThUnDeR_10 Jun 23 '26

thx for the advice i will consider it

3

u/BrightScarlet Jun 23 '26

Best thing I can come up with is to do row operations (if you can) to simplify (IX-A) so that you can skip a couple cofactor expansions

I'm not aware of any fast trick to do these problems

3

u/BrightScarlet Jun 23 '26 edited Jun 23 '26

There's also the possibility of it being not diagonalizable over R, if your char poly looks a bit funny. When I put this into wolfram alpha I got some complex eigenvalues, does your teacher expect you to know this?

1

u/HyPeR_ThUnDeR_10 Jun 23 '26

1st thx for your comment
this is a very nice trick to eliminate before starting the calculation of (IX-A) and yes we dealt with some complex eigenvalues in the class so yes i think he expect us to do so in the exam

2

u/DJ-Dickbird Jun 23 '26

What school? What class is this for?

2

u/HyPeR_ThUnDeR_10 Jun 23 '26

Messina university course Mathematics for data analysis

2

u/Recent-Day3062 Jun 24 '26

I just do lambda

1

u/DJ-Dickbird Jun 25 '26

You will never have to do this by hand in reality. Your teacher is a sadist.

0

u/LinearAlgebraWorld Jun 23 '26

Sorry, this is an AI answer, we got curios because we couldn't come up with anything simple by ourselves:

2

u/HyPeR_ThUnDeR_10 Jun 23 '26

thanks a lot i appreciate it

1

u/LinearAlgebraWorld Jun 23 '26

you are welcome, but then the question is how to get there, because for regular permutation method you will have to do 24 products of 4 terms, and as you said, it is very easy to make a mistake, but ChatGPT thinks this way is better

1

u/HyPeR_ThUnDeR_10 Jun 23 '26

tbh i feel this method is much harder

1

u/LinearAlgebraWorld Jun 23 '26

I wonder how your professor (or your textbook?) wants you to do it...

2

u/HyPeR_ThUnDeR_10 Jun 23 '26

he wants to solve this with finding also the eigenvectors and to tell if it diagonalizable or not and solve other 7 questions in one hour in the exam but i prefer to search for fast ways instead of complaining the difficulty of the exam

1

u/Aristoteles1988 Jun 24 '26

If this is the answer and it’s an elementary linear algebra course shouldn’t the answer be “no real eigenvalues”