Q1: Let $A,B$ matrices given by $A = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix} $ and $B = \begin{pmatrix} 1 & 2 \ 1 & 2 \end{pmatrix}$. What is correct for $AB - BA$?

A1: $AB - BA \neq \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} $

A2: $AB - BA = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} $

A3: $AB - BA = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} $

A4: $AB - BA $ is not well-defined.

Q2: What is the correct associative law for the matrix multiplication, assuming $A \in \mathbb{R}^{m \times n}$, $B \in \mathbb{R}^{n\times k}$, and $C \in \mathbb{R}^{k \times \ell}$?

A1: $ (A B) C = A (BC) $

A2: $ (A + B) C = A C + B C $

A3: $ (A + B) + C = A C + B C $

A4: $ (A B ) C = B ( A C) $

A5: $ A B = BC $

Q3: Let $A,B$ be matrices where $A$ is given by $A = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$. What is the matrix product $AB$?

A1: $B$

A2: $A$

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

A4: $A+B$