To find the matrix [tex]\( P = \begin{pmatrix} p_1 \\ p_2 \end{pmatrix} \)[/tex], we need to solve the system of linear equations given by:
[tex]\[
\begin{pmatrix} -1 & 2 \\ 2 & -2 \end{pmatrix} \begin{pmatrix} p_1 \\ p_2 \end{pmatrix} = \begin{pmatrix} -2 \\ 4 \end{pmatrix}
\][/tex]
This system can be written as two equations:
1. [tex]\(-1 \cdot p_1 + 2 \cdot p_2 = -2\)[/tex]
2. [tex]\(2 \cdot p_1 - 2 \cdot p_2 = 4\)[/tex]
Let's solve these equations step-by-step.
Step 1: Solve the first equation for [tex]\( p_1 \)[/tex]
[tex]\[
-1 \cdot p_1 + 2 \cdot p_2 = -2
\][/tex]
Rearranging for [tex]\( p_1 \)[/tex]:
[tex]\[
-1 \cdot p_1 = -2 - 2 \cdot p_2
\][/tex]
So,
[tex]\[
p_1 = 2 + 2 \cdot p_2
\][/tex]
Step 2: Substitute [tex]\( p_1 \)[/tex] into the second equation
Substitute [tex]\( p_1 = 2 + 2 \cdot p_2 \)[/tex] into the second equation:
[tex]\[
2 \cdot (2 + 2 \cdot p_2) - 2 \cdot p_2 = 4
\][/tex]
Expanding and simplifying:
[tex]\[
4 + 4 \cdot p_2 - 2 \cdot p_2 = 4
\][/tex]
[tex]\[
4 + 2 \cdot p_2 = 4
\][/tex]
Step 3: Solve for [tex]\( p_2 \)[/tex]
Subtract 4 from both sides:
[tex]\[
2 \cdot p_2 = 0
\][/tex]
[tex]\[
p_2 = 0
\][/tex]
Step 4: Solve for [tex]\( p_1 \)[/tex]
Now substitute [tex]\( p_2 = 0 \)[/tex] back into the equation [tex]\( p_1 = 2 + 2 \cdot p_2 \)[/tex]:
[tex]\[
p_1 = 2 + 2 \cdot 0
\][/tex]
[tex]\[
p_1 = 2
\][/tex]
Thus, the matrix [tex]\( P \)[/tex] is:
[tex]\[
P = \begin{pmatrix} 2 \\ 0 \end{pmatrix}
\][/tex]
