JEE Main & Advanced Mathematics Determinants & Matrices Geometrical Transformations

(1) Reflexion in the x-axis: If \[P'\,\,(x',y')\]is the reflexion of the point \[P(x,y)\]on the x-axis, then the matrix \[\left[ \begin{matrix} 1 & 0  \\ 0 & -1  \\\end{matrix} \right]\] describes the reflexion of a point \[P(x,y)\]in the x-axis.

(2) Reflexion in the y-axis 

Here the matrix is \[\left[ \begin{matrix} -1 & 0  \\ 0 & 1  \\\end{matrix} \right]\]

(3) Reflexion through the origin

Here the matrix is \[\left[ \begin{matrix} -1 & 0  \\ 0 & -1  \\ \end{matrix} \right]\]

(4) Reflexion in the line  \[\mathbf{y=x}\]

Here the matrix is \[\left[ \begin{matrix} 0 & 1  \\ 1 & 0  \\ \end{matrix} \right]\]

(5) Reflexion in the line \[\mathbf{y=}-\mathbf{x}\]

Here the matrix is \[\left[ \begin{matrix} \,\,0 & -1  \\ -1 & \,\,0  \\ \end{matrix} \right]\]

(6) Reflexion in \[y=x\,\mathbf{tan\theta }\]

Here matrix is \[\left[ \begin{matrix} \cos 2\theta  & \sin 2\theta   \\ \sin 2\theta  & -\cos 2\theta   \\ \end{matrix} \right]\]

(7) Rotation through an angle \[\mathbf{\theta }\]

Here matrix is \[\left[ \begin{matrix} \cos \theta  & -\sin \theta   \\ \sin \theta  & \cos \theta   \\ \end{matrix} \right]\]