91影视

A transformation Matrix is defined as a matrix that transforms one vector by the process of matrix multiplication.

Let X be the matrix of the transformation T . We have to find the matrix X such that$X{P}_0=P_1,X{P}_1=P_{3}andX{P}_2=P_2$. This system can be written as XY=Z , where

$$\begin{gathered} Y=\left[\begin{array}{ccc}1& 1& -1\\ 1& -1& 1\\ 1& -1& -1\end{array}\right] \\ and \\ Z=\left[\begin{array}{ccc}1& -1& -1\\ -1& -1& 1\\ -1& 1& -1\end{array}\right] \end{gathered}$$

Therefore $X=Z{Y}^{-1}$.First we have to find$Y^{-1}$

Now we will use the formula ${\mathbf{Y}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\frac{\mathbf{adj}\left(Y\right)}{\mathbf{adj}\left(Y\right)}$

$Y^{-1}=\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& 0\\ \frac{1}{2}& 0& -\frac{1}{2}\\ 0& \frac{1}{2}& -\frac{1}{2}\end{array}\right]$

$$\begin{gathered} X=Z{Y}^{-1} \\ X=\left[\begin{array}{ccc}1& -1& -1\\ -1& -1& 1\\ -1& 1& -1\end{array}\right]\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& 0\\ \frac{1}{2}& 0& -\frac{1}{2}\\ 0& \frac{1}{2}& -\frac{1}{2}\end{array}\right] \\ X=\left[\begin{array}{ccc}0& 0& 1\\ -1& 0& 0\\ 0& -1& 0\end{array}\right] \end{gathered}$$

Let R be the matrix of the transformation L. We have to find the matrix R such that$R{P}_0=P_0,R{P}_1=P_2=P_1$. This system can be written as RS=C, where

$$\begin{gathered} S=\left[\begin{array}{ccc}1& 1& -1\\ 1& -1& 1\\ 1& -1& -1\end{array}\right] \\ and \\ C=\left[\begin{array}{ccc}1& -1& 1\\ 1& 1& -1\\ 1& -1& -1\end{array}\right] \end{gathered}$$

Therefore$R=C{S}^{-1}$ . First we have to find$S^{-1}$ .

Now we will use the formula ${\mathbf{S}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\frac{\mathbf{adj}\left(S\right)}{\mathbf{adj}\left(S\right)}$

role="math" localid="1659352879074" $S^{-1}=\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& 0\\ \frac{1}{2}& 0& -\frac{1}{2}\\ 0& \frac{1}{2}& -\frac{1}{2}\end{array}\right]$

$$\begin{gathered} R=C{S}^{-1} \\ R=\left[\begin{array}{ccc}1& -1& 1\\ 1& 1& -1\\ 1& -1& -1\end{array}\right]\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& 0\\ \frac{1}{2}& 0& -\frac{1}{2}\\ 0& \frac{1}{2}& -\frac{1}{2}\end{array}\right] \\ R=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right] \end{gathered}$$

$$\begin{gathered} X=\left[\begin{array}{ccc}0& 0& 1\\ -1& 0& 0\\ 0& -1& 0\end{array}\right] \\ R=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right] \end{gathered}$$