#1
Given $\displaystyle A\in M_{2010}(\mathbb{C}).\text{ Defined }A_k=\{x\in\mathbb{C}^n;Ax=kx\}$
$\displaystyle \text{Let } dim(A_1)=1005\text{ and }dim(A_2)=1005$
Then prove that $\displaystyle dim(A_{\frac{3}{2}})=0, dim(A_{\sqrt{2}})=0, dim(A_{-1})=0$
#2
Suppose $\displaystyle A\in M_n(\mathbb{C})\text{ satisfying }det(A^2-4T)=0$,
Show that -2 or 2 is/are eigen value/s of A.
#3
Given $\displaystyle x_1,x_2,...,x_n; n\in\mathbb{R}^+$
If $\displaystyle A=[a_{ij}]\text{ where }a_{ij}=\sum_{k=1}^n a_k^{i+j-2},\forall i,j\in\{1,2,...,n\}.$
Show that $\displaystyle det(A)\geq 0$
And when det(A) = 0 ?