Question

The 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: In general whenever we take the reflection of a thing than we generally look the object opposite in dimension as the viewing will be in such a way that if a point is taken then we have to take a coordinate which is opposite to the given line in the plane. Hence, for considering the identity matrix we have to take its reflection along the x-axis and y-axis hence we have to look at the coordinates exactly at \[{180^0}\]. An identity matrix is \[I = \left[ {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right]\].

Complete step-by-step answer:
As the given line is \[x + y = 0\] and the identity matrix is \[I = \left[ {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right]\].
So, we now take the identity matrix’s reflection along the x-axis so it will be
\[ = \left[ {\begin{array}{*{20}{c}}
  1&0 \\
  0&{ - 1}
\end{array}} \right]\]
Now, taking the reflection of the identity matrix along the y-axis so it will be,
\[ = \left[ {\begin{array}{*{20}{c}}
  { - 1}&0 \\
  0&1
\end{array}} \right]\]
Hence, from both the above contents we can see that the reflection of the line along the line given \[x + y = 0\] is
\[ = \left[ {\begin{array}{*{20}{c}}
  { - 1}&0 \\
  0&{ - 1}
\end{array}} \right]\]
Hence, the required matrix of the transformation reflection in the line \[x + y = 0\]is \[ = \left[ {\begin{array}{*{20}{c}}
  { - 1}&0 \\
  0&{ - 1}
\end{array}} \right]\]
Hence correct option in C.

Note: When you reflect a point across the line \[y = x\], the x-coordinate, and y-coordinate change places. If you reflect over the line \[y = - x\], the x-coordinate and y-coordinate change places and are negated. The line \[y = x\] is the point \[\left( {y,x} \right)\].