matrix = 21...what it means??

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+3 votes

The condition for which the eigenvalues of the matrix $A=\begin{bmatrix} 2 & 1\\ 1 &k \end{bmatrix}$ are positive is

- $k > \frac{1}{2}$
- $ k > −2$
- $ k > 0$
- $k< \frac{-1}{2}$

0 votes

By the properties of eigen values, if all the principal minors of A are possitive then all the eigen values of A are also possitive.

so |A_{2*2}| > 0

ie 2k - 1 > 0

k>1/2

0

you are taking det(A)>0..

but what if both eigen values are negative then also det(A)>0 will follow..

but what if both eigen values are negative then also det(A)>0 will follow..

+9

We know that Det(A) = product of eigen values and Tr(A) = sum of eigen values. Therefore, to have both eigen values as positive , det(A) should be >0 & Tr(A) >0.So, we can check for both the conditions on the given matrix:

Det(A) > 0 => 2k-1>0 => k>1/2

and Tr(A) > 0 => k+2 > 0 => k>-2

Hence, the first solution satisfies both the condition. So, the answer is A) .. I think this is a better approach than the Best Answer here @Joshi_nitish

Det(A) > 0 => 2k-1>0 => k>1/2

and Tr(A) > 0 => k+2 > 0 => k>-2

Hence, the first solution satisfies both the condition. So, the answer is A) .. I think this is a better approach than the Best Answer here @Joshi_nitish

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