Let's tackle this question step-by-step.
### (a) Write down the matrices of [tex]\( P, Q \)[/tex], and [tex]\( R \)[/tex]
First, we need to translate the given transformations into matrix form.
1. Transformation [tex]\( P \)[/tex]
[tex]\[
P: (x, y) \rightarrow (-4x - y, 2x)
\][/tex]
To represent this transformation as a matrix, we can write:
[tex]\[
\begin{pmatrix}
-4x - y \\
2x
\end{pmatrix}
\][/tex]
This corresponds to the matrix:
[tex]\[
P = \begin{pmatrix}
-4 & -1 \\
2 & 0
\end{pmatrix}
\][/tex]
2. Transformation [tex]\( Q \)[/tex]
[tex]\[
Q: (x, y) \rightarrow (y, 6x - 9y)
\][/tex]
To represent this transformation as a matrix, we can write:
[tex]\[
\begin{pmatrix}
y \\
6x - 9y
\end{pmatrix}
\][/tex]
This corresponds to the matrix:
[tex]\[
Q = \begin{pmatrix}
0 & 1 \\
6 & -9
\end{pmatrix}
\][/tex]
3. Transformation [tex]\( R \)[/tex]
[tex]\[
R: (x, y) \rightarrow (x - 2y, 3x + 5y)
\][/tex]
To represent this transformation as a matrix, we can write:
[tex]\[
\begin{pmatrix}
x - 2y \\
3x + 5y
\end{pmatrix}
\][/tex]
This corresponds to the matrix:
[tex]\[
R = \begin{pmatrix}
1 & -2 \\
3 & 5
\end{pmatrix}
\][/tex]
The matrices for [tex]\( P, Q \)[/tex], and [tex]\( R \)[/tex] are:
[tex]\[ P = \begin{pmatrix}
-4 & -1 \\
2 & 0
\end{pmatrix} \][/tex]
[tex]\[ Q = \begin{pmatrix}
0 & 1 \\
6 & -9
\end{pmatrix} \][/tex]
[tex]\[ R = \begin{pmatrix}
1 & -2 \\
3 & 5
\end{pmatrix} \][/tex]
### (b) Calculate the result matrices
The result obtained for the matrices from the transformations is:
[tex]\[ P = \begin{pmatrix}
-4 & -1 \\
2 & 0
\end{pmatrix} \][/tex]
[tex]\[ Q = \begin{pmatrix}
0 & 1 \\
6 & -9
\end{pmatrix} \][/tex]
[tex]\[ R = \begin{pmatrix}
1 & -2 \\
3 & 5
\end{pmatrix} \][/tex]
Hence, the matrices for these transformations are correctly identified and look like this:
[tex]\[ P = \begin{pmatrix}
-4 & -1 \\
2 & 0
\end{pmatrix} \][/tex]
[tex]\[ Q = \begin{pmatrix}
0 & 1 \\
6 & -9
\end{pmatrix} \][/tex]
[tex]\[ R = \begin{pmatrix}
1 & -2 \\
3 & 5
\end{pmatrix} \][/tex]
