The set of all matrices with trace 1 is path connected.

Question

Problem: Prove that the set of all matrices with trace 1 is path connected.

Approach: Given set $X=\{A\in M_n(\mathbb{C}): Tr(A)=1\}$. We show that $X$ is path connected. Let $\gamma(t):[0,1]\to X$ such that $\gamma(t)=tA+(1-t)B$.Then $\gamma(0)=B$ and $\gamma(1)=A$. We claim that $\gamma$ is a continuous map joining two elements $A$ and $B$.

Define a map $Tr:X\to \mathbb{C}$.

Now, $[0,1]\to^{\gamma} X\to^{Tr}\Bbb{C}$

So the composition map $Tr\circ\gamma[0,1]\to \Bbb{C}$ is continuous and as Trace is linear so:

$Tr\circ \gamma(t)=Tr(\gamma(t))=Tr(tA+(1-t)B)=Tr(A)t+(1-t)Tr(B)=1$.

Hence, $\gamma$ is a required path.

Answer 1

As @cooper.hat verified the answer so I don't want to leave it answered.

Given set $\color{blue}{X=\{A\in M_n(\mathbb{C}): Tr(A)=1\}.}$

We show that $X$ is path connected. Let $\color{green}{\gamma(t):[0,1]\to X}$ be a map defined by $\color{brown}{\gamma(t):=tA+(1-t)B}$. Then $\gamma(0)=B$ and $\gamma(1)=A$. We claim that $\gamma$ is a continuous map joining two elements $A$ and $B$.

Define a map $\color{maroon}{Tr:X\to \mathbb{C}}$.

Now, $\color{blue}{[0,1]\to^{\gamma} X\to^{Tr}\Bbb{C}}$

So the composition map $\color{red}{Tr\circ\gamma:[0,1]\to \Bbb{C}}$ is continuous and as Trace is linear so:

$\color{blue}{Tr\circ \gamma(t)=Tr(\gamma(t))=Tr(tA+(1-t)B)=Tr(A)t+(1-t)Tr(B)=1}$.

Hence, $\gamma$ is a required path.