linear algebra - What subspace of 3 x 3 matrices is spanned by the invertible matrices? Rank 1 matrices? [GStrang P181, 3.5.29(a)(b)]

What subspace of $3$ by $3$ matrices is spanned (take all combinations) by
(a) the invertible matrices?
(b) the rank one matrices?

Answer: (a) The invertible matrices span the space of all $3$ by $3$ matrices.
(b) The rank one matrices also span the space of all $3$ by $3$ matrices. $\quad \square$

P144: The rank of a matrix is its number of pivots.
P171: A set of vectors spans a space if their linear combinations fill the space.

How'd you divine that these matrices fulfill the questions? The answers don't explain.

For (a), I recalled that invertible matrices have $n$ pivots (1 in each row) and so $n$ linearly-independent columns.

(b) Rank one matrices must've only 1 pivot. Thus, its $n - 1$ columns are linearly dependent. Then what?

This question precedes dimensions/theorems of the 4 subspaces, Orthogonality, Determinants, eigenvalues and eigenvectors, and linear transformations. Please pretermit them.

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Supplementary added on Nov 26

The standard basis for $\mathbb{M}_{n \times n}$ is $\{E(i, j)\}_{1 \le i, j\le n}$ with $1$ in the $i$th row and $j$th column and $0$ elsewhere. Call this $S$.

1. Is the next step rewriting, in terms of $S$, all (a) the invertible matrices and (b) rank one matrices.




2. Then, how'd I describe the space of all the invertible matrices (neither a subspace nor a vector space)? Since these two matrices in Deven Ware's last equation have 3 and 2 pivots respectively, they're invertible, but don't span the set of all invertible matrices? $\left( \begin{array}{ccc} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array} \right), \left( \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$




3. I know that each of the $n\cdot n$ matrices of size $n \times n$ in $S$ are rank one, but how'd I describe the set of all rank $1$ matrices?