To find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the equation
[tex]\[
\left(\begin{array}{cc}-1 & 0 \\ 0 & -2\end{array}\right)\binom{x}{y}=\binom{-2}{4},
\][/tex]
we need to solve the matrix equation given by:
[tex]\[
\left(\begin{array}{cc}-1 & 0 \\ 0 & -2\end{array}\right) \left(\begin{array}{c} x \\ y \end{array}\right) = \left(\begin{array}{c} -2 \\ 4 \end{array}\right).
\][/tex]
This results in the following system of linear equations:
1. [tex]\(-1 \cdot x + 0 \cdot y = -2\)[/tex]
2. [tex]\(0 \cdot x - 2 \cdot y = 4\)[/tex]
We can simplify these equations:
1. [tex]\(-x = -2\)[/tex]
2. [tex]\(-2y = 4\)[/tex]
Let's solve these equations one by one.
For the first equation:
[tex]\[
-x = -2
\][/tex]
Multiplying both sides by [tex]\(-1\)[/tex]:
[tex]\[
x = 2
\][/tex]
For the second equation:
[tex]\[
-2y = 4
\][/tex]
Dividing both sides by [tex]\(-2\)[/tex]:
[tex]\[
y = -2
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
\binom{x}{y} = \binom{2}{-2}.
\][/tex]
So, the matrix [tex]\( \binom{x}{y} \)[/tex] is:
[tex]\[
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
2 \\
-2
\end{pmatrix}.
\][/tex]
