Answer :
To solve for matrix [tex]\(X\)[/tex] given that [tex]\(\frac{1}{2} X = A\)[/tex], where [tex]\(A = \left[\begin{array}{cc}6 & -12 \\ 4 & 8\end{array}\right]\)[/tex], we need to find a matrix [tex]\(X\)[/tex] such that when halved, equals matrix [tex]\(A\)[/tex].
Here’s a step-by-step approach:
1. Start with the equation [tex]\(\frac{1}{2} X = A\)[/tex].
2. To clear the fraction, multiply both sides of the equation by 2:
[tex]\[ X = 2A \][/tex]
3. Substitute the given matrix [tex]\(A\)[/tex] into the equation:
[tex]\[ X = 2 \left[\begin{array}{cc}6 & -12 \\ 4 & 8\end{array}\right] \][/tex]
4. Perform scalar multiplication by individually multiplying each element in matrix [tex]\(A\)[/tex] by 2:
[tex]\[ X = \left[\begin{array}{cc} 2 \cdot 6 & 2 \cdot (-12) \\ 2 \cdot 4 & 2 \cdot 8 \end{array}\right] \][/tex]
5. This results in:
[tex]\[ X = \left[\begin{array}{cc}12 & -24 \\ 8 & 16\end{array}\right] \][/tex]
So, matrix [tex]\(X\)[/tex], given [tex]\(\frac{1}{2} X = A\)[/tex], is:
[tex]\[ \left[\begin{array}{cc}12 & -24 \\ 8 & 16\end{array}\right]. \][/tex]
Thus, the correct answer from the options is:
[tex]\[ \left[\begin{array}{cc}12 & -24 \\ 8 & 16\end{array}\right]. \][/tex]
Here’s a step-by-step approach:
1. Start with the equation [tex]\(\frac{1}{2} X = A\)[/tex].
2. To clear the fraction, multiply both sides of the equation by 2:
[tex]\[ X = 2A \][/tex]
3. Substitute the given matrix [tex]\(A\)[/tex] into the equation:
[tex]\[ X = 2 \left[\begin{array}{cc}6 & -12 \\ 4 & 8\end{array}\right] \][/tex]
4. Perform scalar multiplication by individually multiplying each element in matrix [tex]\(A\)[/tex] by 2:
[tex]\[ X = \left[\begin{array}{cc} 2 \cdot 6 & 2 \cdot (-12) \\ 2 \cdot 4 & 2 \cdot 8 \end{array}\right] \][/tex]
5. This results in:
[tex]\[ X = \left[\begin{array}{cc}12 & -24 \\ 8 & 16\end{array}\right] \][/tex]
So, matrix [tex]\(X\)[/tex], given [tex]\(\frac{1}{2} X = A\)[/tex], is:
[tex]\[ \left[\begin{array}{cc}12 & -24 \\ 8 & 16\end{array}\right]. \][/tex]
Thus, the correct answer from the options is:
[tex]\[ \left[\begin{array}{cc}12 & -24 \\ 8 & 16\end{array}\right]. \][/tex]
Answer:
D
Step-by-step explanation:
Given the matrix [tex]A = \left[\begin{array}{cc} 6 & -12 \\ 4 & 8 \end{array}\right][/tex]and the equation [tex]\frac{1}{2} X = A[/tex], we need to find matrix X .
First, let's isolate X in the equation. Multiply both sides of the equation by 2:
X = 2A
Now, let's calculate 2A :
[tex]2 \times \left[\begin{array}{cc} 6 & -12 \\ 4 & 8 \end{array}\right] \\\\ \left[\begin{array}{cc} 2 \times 6 & 2 \times -12 \\ 2 \times 4 & 2 \times 8 \end{array}\right] \\\\\left[\begin{array}{cc} 12 & -24 \\ 8 & 16 \end{array}\right][/tex]
Therefore, the correct matrix X is:
D.[tex]\left[\begin{array}{cc} 12 & -24 \\ 8 & 16 \end{array}\right][/tex]