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

Hint:
Here we will use the matrix for transformation reflection of a straight line directly putting the value of slope m.
Formula used:
\[ \Rightarrow \dfrac{1}{{1 + {m^2}}}\left[ {\begin{array}{*{20}{c}}
  {1 - {m^2}}&{2m} \\
  {2m}&{{m^2} - 1}
\end{array}} \right]\] 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.