Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.

Question

Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.(Take into consideration metric $d_2...$)

I was thinking to show that this is a linear image between normed spaces, but am not sure how to do that because of $x$ and $y$ thereby showing that it is a continuous function. And everyone knows that the continuous map of a compact set is compact. Could it be that $A(x)$ is accually $A(x,y)$? Its very clear on my page that its $A(x).$

Answer 1

The map $x \mapsto A(x)$ is continuous. The interval $[0,1]$ is compact. You are now done. Note that all norms on a finite dimensional space are equivalent.