$GL(n,\mathbb R)$ and $SL(n,\mathbb R)\times \mathbb R^\ast$

Question

I learn this one from Introduction to Manifolds by Loring W. Tu.

I wonder why $GL(n,\mathbb R)$ and $SL(n,\mathbb R) \times \mathbb R^\ast$ are isomorphic iff $n$ is odd. I mislead myself to see that his discussion in (b) also holds for n odd, and I can't tell where is wrong...

Any help will be appreciated.

Answer 1

Hint: Recall that elements of the special linear group must have determinant equal to $1$. Now, consider the following matrices:

$$ \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} $$

$$ \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}$$

Is the first an element of $SL(2,\Bbb R)$? Is the second an element of $SL(3,\Bbb R)$? Can you see how the dimensions of the matrices being odd or even makes a difference here?