Undergrad Is $SO(3)$ Path-Connected?

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Here is this week's problem!

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Show that $SO(3)$, the space of all real $3\times 3$ orthogonal matrices of determinant $1$, is path-connected.
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No one answered this problem. You can read my solution below.

Spoiler

Fix $M\in SO(3)$. There is a factorization $M = QA(\theta)Q^T$ where $Q$ is orthogonal, $A(\theta) = 1 \oplus R(\theta)$ and $R(\theta))$ is a two-dimensional rotation matrix. The map $F : [0,1] \to SO(3)$ defined by $F(t) = QA(t\theta)Q^T$ is a continuous path from the identity matrix to $M$. Hence, $SO(3)$ is path-connected.