Question

The in matrix of the transformation reflection in the line $x + y = 0$ is
A) \left( {\begin{array}{*{20}{c}} 1&0 \\\ 0& {- 1} \end{array}} \right)
B) \left( {\begin{array}{*{20}{c}} 0&1 \\\ 1&0 \end{array}} \right)
C) \left( {\begin{array}{*{20}{c}} {- 1}&0 \\\ 0& {- 1} \end{array}} \right)
D) \left( {\begin{array}{*{20}{c}} 0& {- 1} \\\ {- 1}&0 \end{array}} \right)

Answer 1

Here we will use the matrix for transformation reflection of a straight line directly putting the value of slope m.
Formula used:

{1 - {m^2}}&{2m} \\\ {2m}&{{m^2} - 1} \end{array}} \right]$$ gives the transformation reflection matrix. **Complete step by step solution:** We are given with a line $$x + y = 0$$ i.e. $$y = - x$$ This is an equation of straight line. If compared with the general equation $$y = mx + c$$ we get m=-1. We know that matrix for transformation reflection in the line $$y = mx + c$$ is given by, $$ \Rightarrow \dfrac{1}{{1 + {m^2}}}\left[ {\begin{array}{*{20}{c}} {1 - {m^2}}&{2m} \\\ {2m}&{{m^2} - 1} \end{array}} \right]$$ So let’s substitute the value of m. $$ \Rightarrow \dfrac{1}{{1 + {{( - 1)}^2}}}\left[ {\begin{array}{*{20}{c}} {1 - {{\left( { - 1} \right)}^2}}&{2\left( { - 1} \right)} \\\ {2\left( { - 1} \right)}&{{{\left( { - 1} \right)}^2} - 1} \end{array}} \right]$$ $$ \Rightarrow \dfrac{1}{2}\left[ {\begin{array}{*{20}{c}} 0&{ - 2} \\\ { - 2}&0 \end{array}} \right]$$ Multiplying the matrix terms with half we get $$ \Rightarrow \left[ {\begin{array}{*{20}{c}} 0&{ - 1} \\\ { - 1}&0 \end{array}} \right]$$ And this is our answer. **Hence, option D is correct.** **Note:** Here using the matrix of transformation reflection will help you to get a direct answer. But remember the sign of slope. This is a standard formula and it’s always recommended to remember it.