Sure! Let's go through the steps to solve the matrix equation [tex]\( A \times P = B \)[/tex].
The given matrix equation is:
[tex]\[
\begin{pmatrix} -1 & 2 \\ 2 & -2 \end{pmatrix} \times \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -2 \\ 4 \end{pmatrix}
\][/tex]
We can interpret this as a system of linear equations. Expanding the matrix multiplication on the left-hand side, we get:
[tex]\[
\begin{pmatrix} -1 \cdot x + 2 \cdot y \\ 2 \cdot x - 2 \cdot y \end{pmatrix} = \begin{pmatrix} -2 \\ 4 \end{pmatrix}
\][/tex]
This gives us the following system of linear equations:
1. [tex]\(-x + 2y = -2\)[/tex]
2. [tex]\(2x - 2y = 4\)[/tex]
We will solve this system step by step.
First, let's take the second equation and simplify it:
[tex]\[
2x - 2y = 4
\][/tex]
Divide every term by 2:
[tex]\[
x - y = 2 \quad \text{(Equation 3)}
\][/tex]
Now, we have the simplified system of equations:
1. [tex]\(-x + 2y = -2\)[/tex]
2. [tex]\(x - y = 2\)[/tex]
From Equation 3, we can express [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]:
[tex]\[
x = y + 2 \quad \text{(Equation 4)}
\][/tex]
Next, substitute Equation 4 into the first equation:
[tex]\[
-(y + 2) + 2y = -2
\][/tex]
Simplify the left-hand side:
[tex]\[
-y - 2 + 2y = -2
\][/tex]
Combine like terms:
[tex]\[
y - 2 = -2
\][/tex]
Add 2 to both sides:
[tex]\[
y = 0
\][/tex]
Now that we have [tex]\(y\)[/tex], substitute [tex]\(y = 0\)[/tex] back into Equation 4 to find [tex]\(x\)[/tex]:
[tex]\[
x = 0 + 2
\][/tex]
[tex]\[
x = 2
\][/tex]
Thus, we have [tex]\( x = 2 \)[/tex] and [tex]\( y = 0 \)[/tex].
Therefore, the matrix [tex]\( P \)[/tex] is:
[tex]\[
P = \begin{pmatrix} 2 \\ 0 \end{pmatrix}
\][/tex]
In conclusion, the matrix [tex]\( P \)[/tex] that satisfies the equation [tex]\( A \times P = B \)[/tex] is:
[tex]\[
\begin{pmatrix} 2 \\ 0 \end{pmatrix}
\][/tex]
