To solve for matrix [tex]\( X \)[/tex] in the equation [tex]\( X + U = V \)[/tex], we need to isolate [tex]\( X \)[/tex]. We can do this by subtracting matrix [tex]\( U \)[/tex] from matrix [tex]\( V \)[/tex].
So, the equation [tex]\( X + U = V \)[/tex] becomes:
[tex]\[ X = V - U \][/tex]
Now, we will perform matrix subtraction element-wise using the given matrices [tex]\( U \)[/tex] and [tex]\( V \)[/tex].
Given:
[tex]\[
U = \begin{array}{rrr}
1 & 3 & -5 \\
2 & 14 & 11 \\
-8 & 0 & 5 \\
\end{array}
\][/tex]
[tex]\[
V = \begin{array}{rrr}
13 & -1 & -7 \\
-6 & 1 & 19 \\
0 & 15 & 23 \\
\end{array}
\][/tex]
We will subtract each corresponding element in matrix [tex]\( U \)[/tex] from matrix [tex]\( V \)[/tex].
Step-by-step subtraction:
- For the element in the first row and first column:
[tex]\[
V_{11} - U_{11} = 13 - 1 = 12
\][/tex]
- For the element in the first row and second column:
[tex]\[
V_{12} - U_{12} = -1 - 3 = -4
\][/tex]
- For the element in the first row and third column:
[tex]\[
V_{13} - U_{13} = -7 - (-5) = -7 + 5 = -2
\][/tex]
- For the element in the second row and first column:
[tex]\[
V_{21} - U_{21} = -6 - 2 = -8
\][/tex]
- For the element in the second row and second column:
[tex]\[
V_{22} - U_{22} = 1 - 14 = -13
\][/tex]
- For the element in the second row and third column:
[tex]\[
V_{23} - U_{23} = 19 - 11 = 8
\][/tex]
- For the element in the third row and first column:
[tex]\[
V_{31} - U_{31} = 0 - (-8) = 0 + 8 = 8
\][/tex]
- For the element in the third row and second column:
[tex]\[
V_{32} - U_{32} = 15 - 0 = 15
\][/tex]
- For the element in the third row and third column:
[tex]\[
V_{33} - U_{33} = 23 - 5 = 18
\][/tex]
Putting it all together, we get the resulting matrix [tex]\( X \)[/tex]:
[tex]\[
X = \begin{array}{rrr}
12 & -4 & -2 \\
-8 & -13 & 8 \\
8 & 15 & 18 \\
\end{array}
\][/tex]
Thus, the solution for [tex]\( X \)[/tex] is:
[tex]\[
X = \begin{array}{rrr}
12 & -4 & -2 \\
-8 & -13 & 8 \\
8 & 15 & 18 \\
\end{array}
\][/tex]
