To solve the matrix equation [tex]\(-M + KX = J\)[/tex], we need to find the matrix [tex]\(X\)[/tex].
Given:
[tex]\[
M = \begin{pmatrix}
-4 & 0 \\
4 & -1
\end{pmatrix}, \quad
K = \begin{pmatrix}
1 & 3 \\
4 & 3
\end{pmatrix}, \quad
J = \begin{pmatrix}
13 & 12 \\
23 & 13
\end{pmatrix}
\][/tex]
First, we'll rewrite the equation in a more convenient form. We have:
[tex]\[
-M + KX = J
\][/tex]
Rearrange to isolate [tex]\(KX\)[/tex]:
[tex]\[
KX = J + M
\][/tex]
Now, calculate [tex]\(J + M\)[/tex], the sum of the matrices [tex]\(J\)[/tex] and [tex]\(M\)[/tex]:
[tex]\[
J + M = \begin{pmatrix}
13 & 12 \\
23 & 13
\end{pmatrix} + \begin{pmatrix}
-4 & 0 \\
4 & -1
\end{pmatrix} = \begin{pmatrix}
13 + (-4) & 12 + 0 \\
23 + 4 & 13 + (-1)
\end{pmatrix} = \begin{pmatrix}
9 & 12 \\
27 & 12
\end{pmatrix}
\][/tex]
Next, we need to solve for [tex]\(X\)[/tex] by isolating it. To do that, we need the inverse of matrix [tex]\(K\)[/tex]. The inverse [tex]\(K^{-1}\)[/tex] satisfies:
[tex]\[
KK^{-1} = I
\][/tex]
The inverse of a 2x2 matrix [tex]\(K = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\)[/tex] is given by:
[tex]\[
K^{-1} = \frac{1}{ad - bc} \begin{pmatrix}
d & -b \\
-c & a
\end{pmatrix}
\][/tex]
For our matrix [tex]\(K\)[/tex]:
[tex]\[
K = \begin{pmatrix}
1 & 3 \\
4 & 3
\end{pmatrix}
\][/tex]
where [tex]\(a = 1\)[/tex], [tex]\(b = 3\)[/tex], [tex]\(c = 4\)[/tex], and [tex]\(d = 3\)[/tex]. Now calculate the determinant:
[tex]\[
\det(K) = ad - bc = (1)(3) - (3)(4) = 3 - 12 = -9
\][/tex]
Thus, the inverse [tex]\(K^{-1}\)[/tex] is:
[tex]\[
K^{-1} = \frac{1}{-9} \begin{pmatrix}
3 & -3 \\
-4 & 1
\end{pmatrix} = \begin{pmatrix}
-0.3333 & 0.3333 \\
0.4444 & -0.1111
\end{pmatrix}
\][/tex]
With [tex]\(K^{-1}\)[/tex] calculated, we can now solve for [tex]\(X\)[/tex]:
[tex]\[
X = K^{-1}(J + M)
\][/tex]
Perform the matrix multiplication:
[tex]\[
X = \begin{pmatrix}
-0.3333 & 0.3333 \\
0.4444 & -0.1111
\end{pmatrix} \times \begin{pmatrix}
9 & 12 \\
27 & 12
\end{pmatrix} = \begin{pmatrix}
6 & 0 \\
1 & 4
\end{pmatrix}
\][/tex]
Thus, the solution for the equation [tex]\(-M + KX = J\)[/tex] is:
[tex]\[
X= \begin{pmatrix}
6 & 0 \\
1 & 4
\end{pmatrix}
\][/tex]
