#1
Given A\in M_{2010}(\mathbb{C}).\text{ Defined }A_k=\{x\in\mathbb{C}^n;Ax=kx\}
\text{Let } dim(A_1)=1005\text{ and }dim(A_2)=1005
Then prove that dim(A_{\frac{3}{2}})=0, dim(A_{\sqrt{2}})=0, dim(A_{-1})=0
#2
Suppose 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 x_1,x_2,...,x_n; n\in\mathbb{R}^+
If 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 det(A)\geq 0
And when det(A) = 0 ?