Q1: What is the identity matrix in $\mathbb{R}^{2 \times 3}$?

A1: It does not exist.

A2: $ \pmb1 = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \end{pmatrix}$

A3: $ \pmb1 = \begin{pmatrix} 1 & 0 \ 0 & 1 \ 0 & 0\end{pmatrix}$

A4: $ \pmb1 = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$

A5: $ \pmb1 = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}$

Q2: Let $\pmb1$ be the identity matrix in $\mathbb{R}^{n \times n}$. What is the map $f_{\pmb1}$?

A1: The linear map $x \mapsto x$ for all $x \in \mathbb{R}^n$.

A2: The linear map $x \mapsto 3 x$ for all $x \in \mathbb{R}^n$.

A3: The map $x \mapsto \langle x, x \rangle x$ for all $x \in \mathbb{R}^n$.

Q3: Is the matrix $ \begin{pmatrix} 1 & 1 \ 1 & 0 \end{pmatrix}$ singular?

A1: No, because $ \begin{pmatrix} 0 & 1 \ 1 & -1 \end{pmatrix}$ is the inverse of the matrix.

A2: Yes, there is no inverse matrix.

A3: No, because $ \begin{pmatrix} 1 & 1 \ 1 & 0 \end{pmatrix}$ is the inverse of the matrix.

A4: Yes, all matrices in $\mathbb{R}^{2 \times 2}$ are singular.

A5: One needs more information.