Answer :

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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]

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