How to determine which eigenvalue has multiplicity 2 ?

[sid9221]

Say I have a 4x4 matrix and I know 3 eigenvalues and the 3 corresponding eigenvectors.

Is there a fast way to calculate which one has multiplicity 2 without calculating the characteristic polynomial(too time consuming for a 4x4 matrix) or without determining the dimensions of (A-λ I) for each eigenvalue λ. (Row reducing 3 4x4 matrices is too long during an exam)

 

[sid9221]

LOL i just though of the solution as soon as I typed the question.

I guess I could set up an equation using the definition of trace:

Ʃ(elements of diagnal)=λ1+λ2+λ3+x

hence work out x, which will tell me which eigenvalue is repeated.

 

[tiny-tim]

nice!