GATE CSE 2015 Set 2 | Question: 5

Question

The larger of the two eigenvalues of the matrix $\begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$ is _______.

Answer 1

For finding the Eigen Values of a Matrix we need to build the Characteristic equation which is of the form,
$$A-\lambda I$$
where $A$ is the given Matrix, $\lambda$ is a constant, $I$ is the identity matrix.

We'll have a Linear equation after solving $A-\lambda I,$ which will give us two roots for $\lambda.$

$(4 - \lambda) (1 - \lambda) -10 = 0$
$\implies 4 - 5 \lambda+\lambda^2 = 10$
$\implies\lambda^2 - 5\lambda -6 = 0$
$\implies (\lambda - 6)(\lambda +1) = 0$
$\implies \lambda = -1,6.$

$6$ is larger and hence the required answer.

Comments

$\lambda^{2}-(\text{trace of the matrix})\lambda \: + \mid A \mid = 0 $

$\implies \lambda^{2}-5\lambda -6 = 0$

$\implies \lambda = -1,6$

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Perfecto!!!

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Okay i might be the only idiot who instead of focusing on accuracy, just did this quickly and factored it as

X^2-5X-6 =0

(X-3)(X-2)=0

X=2,3

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No no you are not alone. I just did the same mistake.
@SomeEarth

Answer 2

Sum of eigen values= Trace of matrix= 4+1=5

Product of eigen values= Determinant= 4*1-5*2=-6

We can easily find two eigen values using results above

6 and -1.

Comments

I got confused: made it clear :)

The trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal.

https://en.wikipedia.org/wiki/Trace_(linear_algebra)

The sum of the n eigenvalues of A is the same as the trace of A (that is, the sum of the diagonal elements of A).

https://www.adelaide.edu.au/mathslearning/play/seminars/evalue-magic-tricks-handout.pdf

Answer 3

Answer: 6

The two eigen values are of the given matrix are -1 and 6.