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Sagot :
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]
### (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]
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