Re: Matrične jednadžbe
Poslato: Petak, 22. Novembar 2013, 11:12
Odrediti rješenje jednadžbe
[dispmath]AX^{-1}B+AX^{-1}=C^TB^2[/dispmath]
[dispmath]A=\left[\begin{matrix}
2 & 5 \\
1 & 3
\end{matrix}\right],\;B=\left[\begin{matrix}
1 & 2 \\
0 & 1
\end{matrix}\right],\;C=\left[\begin{matrix}
2 & -3 \\
1 & -2
\end{matrix}\right][/dispmath]
Nakon sređivanja dobijem [dispmath]X=(B+I)\left(A^{-1}C^TB^2\right)^{-1}[/dispmath]
[dispmath]B+I=\left[\begin{matrix}
2 & 2 \\
0 & 2
\end{matrix}\right][/dispmath]
[dispmath]C^T=\left[\begin{matrix}
2 & 1 \\
-3 & -2
\end{matrix}\right][/dispmath]
[dispmath]B^2=\left[\begin{matrix}
1 & 4 \\
0 & 1
\end{matrix}\right][/dispmath]
[dispmath]A^{-1}=\left[\begin{matrix}
3 & -5 \\
-1 & 2
\end{matrix}\right][/dispmath]
[dispmath]A^{-1}C^T=\left[\begin{matrix}
21 & 13 \\
-8 & -5
\end{matrix}\right][/dispmath]
[dispmath]A^{-1}C^TB^2=\left[\begin{matrix}
21 & 97 \\
-8 & -37
\end{matrix}\right][/dispmath]
[dispmath]\left(A^{-1}C^TB^2\right)^{-1}=\left[\begin{matrix}
37 & -97 \\
8 & 21
\end{matrix}\right][/dispmath]
[dispmath]X=\left[\begin{matrix}
2 & 1 \\
-3 & -2
\end{matrix}\right]\cdot\left[\begin{matrix}
37 & -97 \\
8 & 21
\end{matrix}\right]=\left[\begin{matrix}
82 & -173 \\
-127 & 249
\end{matrix}\right][/dispmath]
Sumnjam da je točno rješenje
[dispmath]AX^{-1}B+AX^{-1}=C^TB^2[/dispmath]
[dispmath]A=\left[\begin{matrix}
2 & 5 \\
1 & 3
\end{matrix}\right],\;B=\left[\begin{matrix}
1 & 2 \\
0 & 1
\end{matrix}\right],\;C=\left[\begin{matrix}
2 & -3 \\
1 & -2
\end{matrix}\right][/dispmath]
Nakon sređivanja dobijem [dispmath]X=(B+I)\left(A^{-1}C^TB^2\right)^{-1}[/dispmath]
[dispmath]B+I=\left[\begin{matrix}
2 & 2 \\
0 & 2
\end{matrix}\right][/dispmath]
[dispmath]C^T=\left[\begin{matrix}
2 & 1 \\
-3 & -2
\end{matrix}\right][/dispmath]
[dispmath]B^2=\left[\begin{matrix}
1 & 4 \\
0 & 1
\end{matrix}\right][/dispmath]
[dispmath]A^{-1}=\left[\begin{matrix}
3 & -5 \\
-1 & 2
\end{matrix}\right][/dispmath]
[dispmath]A^{-1}C^T=\left[\begin{matrix}
21 & 13 \\
-8 & -5
\end{matrix}\right][/dispmath]
[dispmath]A^{-1}C^TB^2=\left[\begin{matrix}
21 & 97 \\
-8 & -37
\end{matrix}\right][/dispmath]
[dispmath]\left(A^{-1}C^TB^2\right)^{-1}=\left[\begin{matrix}
37 & -97 \\
8 & 21
\end{matrix}\right][/dispmath]
[dispmath]X=\left[\begin{matrix}
2 & 1 \\
-3 & -2
\end{matrix}\right]\cdot\left[\begin{matrix}
37 & -97 \\
8 & 21
\end{matrix}\right]=\left[\begin{matrix}
82 & -173 \\
-127 & 249
\end{matrix}\right][/dispmath]
Sumnjam da je točno rješenje