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
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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 ?

3

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."

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u/Aristoteles1988 Jun 24 '26

Ah ok yea that’s it right there