Uma matriz quadrada Anxn é dita ORTOGONAL se 
= 
=
. Verifique se a matriz M abaixo é ortogonal.
![A^{t}.A \left[\begin{array}{ccc}cosx&-senx&0\\senx&cosx&0\\0&0&1\end{array}\right]](https://tex.z-dn.net/?f=+A%5E%7Bt%7D.A++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dcosx%26amp%3B-senx%26amp%3B0%5C%5Csenx%26amp%3Bcosx%26amp%3B0%5C%5C0%26amp%3B0%26amp%3B1%5Cend%7Barray%7D%5Cright%5D++ "A^{t}.A \left[\begin{array}{ccc}cosx&-senx&0\\senx&cosx&0\\0&0&1\end{array}\right]")
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É dada a matriz M a seguir:
![M = \left[\begin{matrix}\cos x&-\sin x&0\\\sin x &\cos x &0\\0&0&1\end{matrix}\right]](https://tex.z-dn.net/?f=M+%3D+%5Cleft%5B%5Cbegin%7Bmatrix%7D%5Ccos+x%26amp%3B-%5Csin+x%26amp%3B0%5C%5C%5Csin+x+%26amp%3B%5Ccos+x+%26amp%3B0%5C%5C0%26amp%3B0%26amp%3B1%5Cend%7Bmatrix%7D%5Cright%5D "M = \left[\begin{matrix}\cos x&-\sin x&0\\\sin x &\cos x &0\\0&0&1\end{matrix}\right]")
Pede-se para que se verifique se M é uma matriz ortogonal, isto é, se
(o que significa que a transposta de M é a sua inversa).
Primeiramente, vamos escrever a transposta de M:
![M^T=\left[\begin{matrix}\cos x&-\sin x&0\\\sin x &\cos x &0\\0&0&1\end{matrix}\right]^T\\\\\\
M^T=\left[\begin{matrix}\cos x&\sin x&0\\-\sin x &\cos x &0\\0&0&1\end{matrix}\right]](https://tex.z-dn.net/?f=M%5ET%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D%5Ccos+x%26amp%3B-%5Csin+x%26amp%3B0%5C%5C%5Csin+x+%26amp%3B%5Ccos+x+%26amp%3B0%5C%5C0%26amp%3B0%26amp%3B1%5Cend%7Bmatrix%7D%5Cright%5D%5ET%5C%5C%5C%5C%5C%5C%0AM%5ET%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D%5Ccos+x%26amp%3B%5Csin+x%26amp%3B0%5C%5C-%5Csin+x+%26amp%3B%5Ccos+x+%26amp%3B0%5C%5C0%26amp%3B0%26amp%3B1%5Cend%7Bmatrix%7D%5Cright%5D "M^T=\left[\begin{matrix}\cos x&-\sin x&0\\\sin x &\cos x &0\\0&0&1\end{matrix}\right]^T\\\\\\
M^T=\left[\begin{matrix}\cos x&\sin x&0\\-\sin x &\cos x &0\\0&0&1\end{matrix}\right]")
Agora, basta verificarmos o resultado do produto
:
![MM^T=\left[\begin{matrix}\cos x&-\sin x&0\\\sin x &\cos x &0\\0&0&1\end{matrix}\right]\cdot\left[\begin{matrix}\cos x&\sin x&0\\-\sin x &\cos x &0\\0&0&1\end{matrix}\right]=\\\\\\
=\left[\begin{matrix}\cos^2x+\sin^2x+0 &\cos x\sin x-\sin x\cos x+0&0+0+0\\\sin x\cos x-\sin x\cos x+0 & \sin^2x+\cos^2x+0&0+0+0\\0+0+0&0+0+0&0+0+1\end{matrix}\right]\\\\\\
=\left[\begin{matrix}1&0& 0\\0 & 1&0\\0&0&1\end{matrix}\right]\\\\\\
\Longrightarrow \boxed{MM^T=I}](https://tex.z-dn.net/?f=MM%5ET%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D%5Ccos+x%26amp%3B-%5Csin+x%26amp%3B0%5C%5C%5Csin+x+%26amp%3B%5Ccos+x+%26amp%3B0%5C%5C0%26amp%3B0%26amp%3B1%5Cend%7Bmatrix%7D%5Cright%5D%5Ccdot%5Cleft%5B%5Cbegin%7Bmatrix%7D%5Ccos+x%26amp%3B%5Csin+x%26amp%3B0%5C%5C-%5Csin+x+%26amp%3B%5Ccos+x+%26amp%3B0%5C%5C0%26amp%3B0%26amp%3B1%5Cend%7Bmatrix%7D%5Cright%5D%3D%5C%5C%5C%5C%5C%5C%0A%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D%5Ccos%5E2x%2B%5Csin%5E2x%2B0+%26amp%3B%5Ccos+x%5Csin+x-%5Csin+x%5Ccos+x%2B0%26amp%3B0%2B0%2B0%5C%5C%5Csin+x%5Ccos+x-%5Csin+x%5Ccos+x%2B0+%26amp%3B+%5Csin%5E2x%2B%5Ccos%5E2x%2B0%26amp%3B0%2B0%2B0%5C%5C0%2B0%2B0%26amp%3B0%2B0%2B0%26amp%3B0%2B0%2B1%5Cend%7Bmatrix%7D%5Cright%5D%5C%5C%5C%5C%5C%5C%0A%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D1%26amp%3B0%26amp%3B+0%5C%5C0+%26amp%3B+1%26amp%3B0%5C%5C0%26amp%3B0%26amp%3B1%5Cend%7Bmatrix%7D%5Cright%5D%5C%5C%5C%5C%5C%5C%0A%5CLongrightarrow+%5Cboxed%7BMM%5ET%3DI%7D "MM^T=\left[\begin{matrix}\cos x&-\sin x&0\\\sin x &\cos x &0\\0&0&1\end{matrix}\right]\cdot\left[\begin{matrix}\cos x&\sin x&0\\-\sin x &\cos x &0\\0&0&1\end{matrix}\right]=\\\\\\
=\left[\begin{matrix}\cos^2x+\sin^2x+0 &\cos x\sin x-\sin x\cos x+0&0+0+0\\\sin x\cos x-\sin x\cos x+0 & \sin^2x+\cos^2x+0&0+0+0\\0+0+0&0+0+0&0+0+1\end{matrix}\right]\\\\\\
=\left[\begin{matrix}1&0& 0\\0 & 1&0\\0&0&1\end{matrix}\right]\\\\\\
\Longrightarrow \boxed{MM^T=I}")
Portanto, a matriz M é ortogonal.
Pede-se para que se verifique se M é uma matriz ortogonal, isto é, se
Primeiramente, vamos escrever a transposta de M:
Agora, basta verificarmos o resultado do produto
Portanto, a matriz M é ortogonal.
Dani76561:
muito obrigada
De nada!
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