Question: Let \(M_{2,2}\) be the vector space of all \(2 \times 2\) matrices with real entries. We will identify \(M_{2,2}\) with the space \(\mathbb{R}^{4}\) using the basis

\[\left\{\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right) \cdot\right\}\]

By using Implicit Function Theorem show that the set of \(2 \times 2\) -matrices with the determinant equal to 1 is a 3-dimensional smooth manifold in \(M_{2,2}\).

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