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(c) Three linear transformations [tex]\(P, Q\)[/tex], and [tex]\(R\)[/tex] in the [tex]\(xy\)[/tex] plane are defined by:

[tex]\[
\begin{array}{l}
P:(x, y) \rightarrow(-4x - y, 2x) \\
Q:(x, y) \rightarrow(y, 6x - 9y) \\
R:(x, y) \rightarrow(x - 2y, 3x + 5y)
\end{array}
\][/tex]

a) Write down the matrices of [tex]\(P, Q\)[/tex], and [tex]\(R\)[/tex].

b) Find:


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]