Q1: Let $V, W$ be two finite-dimensional $\mathbb{F}$-vector spaces, and $\ell: V \rightarrow W$, $k: W \rightarrow V$ linear maps. What is always correct about a matrix representation of $k \circ \ell$?

A1: It’s a square matrix.

A2: It’s an invertible matrix.

A3: It’s an orthogonal matrix.

A4: It’s a unitary matrix.

Q2: Let $V,W$ be two $\mathbb{F}$-vector spaces of dimension $2$, and $\ell: V \rightarrow W$, $k: W \rightarrow V$ linear maps, with matrix representations $\begin{pmatrix} 2 & 1 \ 3 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$, respectively. What is always correct about a matrix representation of $k \circ \ell$?

A1: It’s given by $\begin{pmatrix} 2 & 1 \ 3 & 1 \end{pmatrix}$.

A2: It’s given by $\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$.

A3: It’s given by $\begin{pmatrix} 1 & 1 \ 1 & 1 \end{pmatrix}$.

A4: It’s an invertible matrix.

Q3: Let $V$ be an $\mathbb{F}$-vector spaces of dimension $2$ and $\ell: V \rightarrow V$ be a linear map with matrix representation $\begin{pmatrix} 2 & 5 \ 1 & 2 \end{pmatrix}$. What is a matrix representation of $ \ell^{-1} $?

A1: $\begin{pmatrix} 2 & 1 \ 2 & 1 \end{pmatrix}$.

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

A3: $\begin{pmatrix} 1 & 1 \ 1 & 1 \end{pmatrix}$.

A4: Every $2\times2$-matrix can be a valid matrix representation of $ \ell^{-1} $.

A5: $\begin{pmatrix} -2 & 5 \ 1 & -2 \end{pmatrix}$.