[latex]
\[
\textbf{Q1: Divide the matrix } A = \begin{pmatrix} 6 & 3 \\ 9 & 12 \end{pmatrix} \textbf{ by the matrix } B = \begin{pmatrix} 3 & 1 \\ 3 & 4 \end{pmatrix}.
\]
\[
\textbf{Solution:}
\]
\[
\textbf{Step 1: Understand matrix division.}
\]
\[
\text{Matrix division } (A \div B) \text{ is interpreted as } A \cdot B^{-1}, \text{ where } B^{-1} \text{ is the inverse of } B.
\]
\[
\textbf{Step 2: Compute the inverse of } B.
\]
\[
B = \begin{pmatrix} 3 & 1 \\ 3 & 4 \end{pmatrix}
\]
\[
\text{The inverse is computed as: } B^{-1} = \frac{1}{\text{det}(B)} \cdot \begin{pmatrix} d & -b \\ -c & a \end{pmatrix},
\]
\[
\text{where } \text{det}(B) = (ad – bc).
\]
\[
\text{For } B, \text{det}(B) = (3 \cdot 4 – 3 \cdot 1) = 9.
\]
\[
B^{-1} = \frac{1}{9} \cdot \begin{pmatrix} 4 & -1 \\ -3 & 3 \end{pmatrix} = \begin{pmatrix} \frac{4}{9} & -\frac{1}{9} \\ -\frac{3}{9} & \frac{3}{9} \end{pmatrix}.
\]
\[
\textbf{Step 3: Multiply } A \text{ by } B^{-1}.
\]
\[
A = \begin{pmatrix} 6 & 3 \\ 9 & 12 \end{pmatrix}, \quad B^{-1} = \begin{pmatrix} \frac{4}{9} & -\frac{1}{9} \\ -\frac{3}{9} & \frac{3}{9} \end{pmatrix}.
\]
\[
A \cdot B^{-1} = \begin{pmatrix} 6 & 3 \\ 9 & 12 \end{pmatrix} \cdot \begin{pmatrix} \frac{4}{9} & -\frac{1}{9} \\ -\frac{3}{9} & \frac{3}{9} \end{pmatrix}.
\]
\[
\textbf{Step 4: Perform the matrix multiplication.}
\]
\[
A \cdot B^{-1} = \begin{pmatrix} (6 \cdot \frac{4}{9} + 3 \cdot -\frac{3}{9}) & (6 \cdot -\frac{1}{9} + 3 \cdot \frac{3}{9}) \\ (9 \cdot \frac{4}{9} + 12 \cdot -\frac{3}{9}) & (9 \cdot -\frac{1}{9} + 12 \cdot \frac{3}{9}) \end{pmatrix}.
\]