Let $n \geqslant 1$ be an integer, and let $M(n)$ denote the set of $n \times n$ real-valued matrices. We make $M(n)$ into an $n^{2}$-dimensional smooth manifold via the obvious identification $M(n)=\mathbb{R}^{n^{2}}$.

(a) Let $G L(n)$ denote the subset

$$G L(n)=\left\{A \in M(n): A^{-1} \text { exists }\right\}$$

Show that $G L(n)$ is a submanifold of $M(n)$. What is $\operatorname{dim} G L(n)$ ?

(b) Now let $S L(n) \subset G L(n)$ denote the subset

$$S L(n)=\{A \in G L(n): \operatorname{det} A=1\}$$

Show that for $A \in G L(n)$,

$$(d \text { det })_{A} B=\operatorname{tr}\left(A^{-1} B\right) \operatorname{det} A .$$

Show that $S L(n)$ is a submanifold of $G L(n)$. What is the dimension of $S L(n) ?$

(c) Now consider the set $X=M(n) \backslash G L(n)$. For what values of $n \geqslant 1$ is $X$ a submanifold of $M(n)$ ?