To solve the equation [tex]\(-M + KX = J\)[/tex] for [tex]\(X\)[/tex], follow these steps:
1. Write down the given matrices:
[tex]\[
M = \begin{bmatrix}
-4 & 0 \\
4 & -1
\end{bmatrix}, \quad K = \begin{bmatrix}
1 & 3 \\
4 & 3
\end{bmatrix}, \quad J = \begin{bmatrix}
13 & 12 \\
23 & 13
\end{bmatrix}
\][/tex]
2. Rearrange the equation:
The given equation is [tex]\(-M + KX = J\)[/tex]. To isolate [tex]\(KX\)[/tex], add matrix [tex]\(M\)[/tex] to both sides:
[tex]\[
KX = J + M
\][/tex]
3. Calculate [tex]\(J + M\)[/tex]:
Add the matrices [tex]\(J\)[/tex] and [tex]\(M\)[/tex]:
[tex]\[
J + M = \begin{bmatrix}
13 & 12 \\
23 & 13
\end{bmatrix} + \begin{bmatrix}
-4 & 0 \\
4 & -1
\end{bmatrix} = \begin{bmatrix}
13 + (-4) & 12 + 0 \\
23 + 4 & 13 + (-1)
\end{bmatrix} = \begin{bmatrix}
9 & 12 \\
27 & 12
\end{bmatrix}
\][/tex]
4. Find the inverse of [tex]\(K\)[/tex]:
The next step is to solve for [tex]\(X\)[/tex] by isolating it on one side. For this, multiply both sides by [tex]\(K^{-1}\)[/tex], where [tex]\(K^{-1}\)[/tex] is the inverse of matrix [tex]\(K\)[/tex]. The inverse of [tex]\(K\)[/tex] is given by:
[tex]\[
K^{-1} = \begin{bmatrix}
-0.33333333 & 0.33333333 \\
0.44444444 & -0.11111111
\end{bmatrix}
\][/tex]
5. Multiply [tex]\(K^{-1}\)[/tex] with [tex]\(J + M\)[/tex] to find [tex]\(X\)[/tex]:
Use the calculated inverse of [tex]\(K\)[/tex] to find [tex]\(X\)[/tex]:
[tex]\[
X = K^{-1}(J + M) = \begin{bmatrix}
-0.33333333 & 0.33333333 \\
0.44444444 & -0.11111111
\end{bmatrix} \times \begin{bmatrix}
9 & 12 \\
27 & 12
\end{bmatrix}
\][/tex]
Perform the matrix multiplication:
[tex]\[
X = \begin{bmatrix}
(-0.33333333 \cdot 9 + 0.33333333 \cdot 27) & (-0.33333333 \cdot 12 + 0.33333333 \cdot 12) \\
(0.44444444 \cdot 9 + -0.11111111 \cdot 27) & (0.44444444 \cdot 12 + -0.11111111 \cdot 12)
\end{bmatrix}
\][/tex]
Simplifying each element:
[tex]\[
X = \begin{bmatrix}
6 & -2.22044605 \cdot 10^{-16} \\
1 & 4
\end{bmatrix}
\][/tex]
Therefore, the solution to the equation [tex]\(-M + KX = J\)[/tex] is:
[tex]\[
X = \begin{bmatrix}
6 & -2.22044605 \cdot 10^{-16} \\
1 & 4
\end{bmatrix}
\][/tex]
