To solve for the matrix [tex]\( X \)[/tex] such that [tex]\( B - X = A \)[/tex], we rearrange the equation to solve for [tex]\( X \)[/tex]:
[tex]\[
X = B - A
\][/tex]
Given the matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
A = \begin{pmatrix}
-1 & -2 & 3 \\
4 & 8 & -6 \\
0 & 1 & 5
\end{pmatrix}
\][/tex]
[tex]\[
B = \begin{pmatrix}
-5 & -1 & 6 \\
4 & 1 & 2 \\
0 & -3 & 2
\end{pmatrix}
\][/tex]
We perform the matrix subtraction element-wise:
[tex]\[
X = B - A
\][/tex]
[tex]\[
X = \begin{pmatrix}
-5 & -1 & 6 \\
4 & 1 & 2 \\
0 & -3 & 2
\end{pmatrix}
- \begin{pmatrix}
-1 & -2 & 3 \\
4 & 8 & -6 \\
0 & 1 & 5
\end{pmatrix}
\][/tex]
Now, subtract each corresponding element:
For the first row:
[tex]\[
\begin{pmatrix}
-5 - (-1) & -1 - (-2) & 6 - 3
\end{pmatrix} = \begin{pmatrix}
-4 & 1 & 3
\end{pmatrix}
\][/tex]
For the second row:
[tex]\[
\begin{pmatrix}
4 - 4 & 1 - 8 & 2 - (-6)
\end{pmatrix} = \begin{pmatrix}
0 & -7 & 8
\end{pmatrix}
\][/tex]
For the third row:
[tex]\[
\begin{pmatrix}
0 - 0 & -3 - 1 & 2 - 5
\end{pmatrix} = \begin{pmatrix}
0 & -4 & -3
\end{pmatrix}
\][/tex]
Combining these results, we get the matrix [tex]\( X \)[/tex]:
[tex]\[
X = \begin{pmatrix}
-4 & 1 & 3 \\
0 & -7 & 8 \\
0 & -4 & -3
\end{pmatrix}
\][/tex]
Therefore, the correct option is:
[tex]\[
\begin{pmatrix}
-4 & 1 & 3 \\
0 & -7 & 8 \\
0 & -4 & -3
\end{pmatrix}
\][/tex]
So, the correct answer is:
[tex]\[
\left[\begin{array}{ccc}-4 & 1 & 3 \\ 0 & -7 & 8 \\ 0 & -4 & -3\end{array}\right]
\][/tex]
