### Given Matrices
We have two matrices:
[tex]\[
A = \begin{pmatrix}
3 & 2 \\
6 & 1
\end{pmatrix}
\][/tex]
and
[tex]\[
M = \begin{pmatrix}
1 & 5 \\
3 & -5 \\
-4 & 9
\end{pmatrix}
\][/tex]
### Steps to Solve
Since matrix multiplication is not directly possible between these matrices due to incompatible dimensions (A is a 2x2 matrix and M is a 3x2 matrix), and we observe that matrix A is referenced with a bizarre notational confusion, we will evaluate the plausible results based on standard matrix operations available.
#### Part (i) and (iii) : [tex]\( AM \)[/tex]
Matrix multiplication requires that the number of columns in the first matrix be equal to the number of rows in the second matrix. Hence, here, [tex]\( AM \)[/tex] directly is not computable because of their incompatible dimension (2x2 with 3x2).
#### Correct Matrices Multiplication: [tex]\( MA \)[/tex]
For matrix multiplication of [tex]\( MA \)[/tex] :
1. The dimension of matrix [tex]\( M \)[/tex] (3x2) and matrix [tex]\( A \)[/tex] (2x2) enables their multiplication.
2. The resultant matrix will have the dimension (3x2).
Let's calculate [tex]\( MA \)[/tex]:
[tex]\[
M = \begin{pmatrix}
1 & 5 \\
3 & -5 \\
-4 & 9
\end{pmatrix}
\][/tex]
[tex]\[
A = \begin{pmatrix}
3 & 2 \\
6 & 1
\end{pmatrix}
\][/tex]
Step-by-step calculation:
[tex]\(
MA = M \cdot A
\)[/tex]
[tex]\[
\begin{pmatrix}
1 \cdot 3 + 5 \cdot 6 & 1 \cdot 2 + 5 \cdot 1 \\
3 \cdot 3 + (-5) \cdot 6 & 3 \cdot 2 + (-5) \cdot 1 \\
-4 \cdot 3 + 9 \cdot 6 & -4 \cdot 2 + 9 \cdot 1
\end{pmatrix}
=\begin{pmatrix}
3 + 30 & 2 + 5 \\
9 - 30 & 6 - 5 \\
-12 + 54 & -8 + 9
\end{pmatrix}
= \begin{pmatrix}
33 & 7 \\
-21 & 1 \\
42 & 1
\end{pmatrix}
\][/tex]
Thus, the resulting matrix after multiplying [tex]\( M \)[/tex] and [tex]\( A \)[/tex] together is:
[tex]\[
\boxed{
MA = \begin{pmatrix}
33 & 7 \\
-21 & 1 \\
42 & 1
\end{pmatrix}
}
\][/tex]
So,
- For part (i), since [tex]\( AM \)[/tex] is not computable directly, we concluded and computed: [tex]\( MA \)[/tex]
- For part (iii), it repeats the question twice without an explicit specified matrix B. Thus irrelevant in this transformed context.
Therefore the valid resultant answer, upholding matrices multiplications can be encapsulated [tex]\( MA \)[/tex] computed above. The numeric matrix computed forms: [tex]\( \begin{pmatrix}33 & 7 \\ -21 & 1 \\ 42 & 1 \end{pmatrix} \)[/tex]
This computational matrix steps and approach leads to correct matrix results.
