MHB Eigenvector Calculations and Verification for y1+y2=5

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The discussion confirms the calculation of eigenvalues for the matrix associated with the equation y1 + y2 = 5. The characteristic polynomial is derived as det(yI - A) = 0, leading to the factors (y-1)(y-4) = 0. The eigenvalues are verified to sum to 5, consistent with the trace of the matrix. Additionally, the use of properties of triangular matrices is suggested as a more straightforward method for such calculations. The results are confirmed by W|A, validating the findings.
ertagon2
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Can you please confirm my answer?
View attachment 7539
det(yI-A) = 0
(y-1)(y-4)=0
y1+y2=5
 

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Yes, it is correct.

Incidentally, did you already learn about the eigenvalues of triangular matrices?
Did you perhaps already learn about the trace of a matrix (= the sum along the main diagonal)?

This could help you to calculate the required quantity without having to set up the characteristic polynomial. (Of course, that is not much work here, either.)
 
I get the characteristic equation:

$$x^2-(1+4)x+(1\cdot4-0\cdot3)=x^2-5x+4=(x-1)(x-4)=0$$

Hence:

$$\lambda_1+\lambda_2=1+4=5\quad\checkmark$$

W|A confirms our results:

W|A - eigenvalues for ((1,0),(3,4))
 
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