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Given matrix [tex]\( Y = \begin{pmatrix} 4 & 8 \\ -3 & -6 \end{pmatrix} \)[/tex] and matrix [tex]\( Z = \begin{pmatrix} -1 & -2 \\ 7 & 2 \end{pmatrix} \)[/tex].

What is matrix [tex]\( X \)[/tex] if [tex]\( \frac{1}{2}(X + Z) = Y \)[/tex]?

A. [tex]\( \begin{pmatrix} 2.5 & 5 \\ -5 & -4 \end{pmatrix} \)[/tex]
B. [tex]\( \begin{pmatrix} 3 & 6 \\ -8.5 & -5 \end{pmatrix} \)[/tex]
C. [tex]\( \begin{pmatrix} 9 & 18 \\ -13 & -14 \end{pmatrix} \)[/tex]



Answer :

GinnyAnswer
To solve for matrix [tex]\( X \)[/tex] in the equation [tex]\(\frac{1}{2}(X + Z) = Y\)[/tex], follow these steps:

1. Start with the given equation:
[tex]\[\frac{1}{2}(X + Z) = Y\][/tex]

2. Eliminate the fraction by multiplying both sides of the equation by 2:
[tex]\[X + Z = 2Y\][/tex]

3. Solve for [tex]\( X \)[/tex] by isolating it on one side of the equation:
[tex]\[X = 2Y - Z\][/tex]

4. Substitute the given matrices [tex]\( Y \)[/tex] and [tex]\( Z \)[/tex] into the equation:
[tex]\[Y = \left[\begin{array}{cc}4 & 8 \\ -3 & -6\end{array}\right]\][/tex]
[tex]\[Z = \left[\begin{array}{cc}-1 & -2 \\ 7 & 2\end{array}\right]\][/tex]

5. Compute [tex]\( 2Y \)[/tex] by multiplying each element of [tex]\( Y \)[/tex] by 2:
[tex]\[2Y = 2 \times \left[\begin{array}{cc}4 & 8 \\ -3 & -6\end{array}\right] = \left[\begin{array}{cc}8 & 16 \\ -6 & -12\end{array}\right]\][/tex]

6. Subtract matrix [tex]\( Z \)[/tex] from [tex]\( 2Y \)[/tex] to find [tex]\( X \)[/tex]:
[tex]\[X = \left[\begin{array}{cc}8 & 16 \\ -6 & -12\end{array}\right] - \left[\begin{array}{cc}-1 & -2 \\ 7 & 2\end{array}\right] = \left[\begin{array}{cc}8+1 & 16+2 \\ -6-7 & -12-2\end{array}\right] = \left[\begin{array}{cc}9 & 18 \\ -13 & -14\end{array}\right]\][/tex]

Therefore, matrix [tex]\( X \)[/tex] is:
[tex]\[ \left[\begin{array}{cc}9 & 18 \\ -13 & -14\end{array}\right] \][/tex]

So the correct option is:
[tex]\[ \left[\begin{array}{cc}9 & 18 \\ -13 & -14\end{array}\right] \][/tex]

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