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

In general whenever we take the reflection of a thing than we generally look the object completely 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 completely opposite to the given line in the plane. Hence, for considering the identity matrix we have take it’s reflection along x-axis and y-axis hence we have to look at the coordinates exactly at ${180^0}$ . Remember the concept that for line $y = mx$the reflection of matrix in general form can be given as A = \dfrac{1}{{1 + {m^2}}}\left[ {\begin{array}{*{20}{c}} {1 - {m^2}}&{2m} \\\ {2m}&{{m^2} - 1} \end{array}} \right] . Hence, rearrange the above given equation and calculate the slope and put the value in the above given matrix.

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] .
Equation of line can also be given as $y = - x$ and hence the value of slope is $m = - 1$.
Now put the value in matrix A = \dfrac{1}{{1 + {m^2}}}\left[ {\begin{array}{*{20}{c}} {1 - {m^2}}&{2m} \\\ {2m}&{{m^2} - 1} \end{array}} \right]
On putting the value of m we get,
$\Rightarrow$ A = \dfrac{1}{2}\left[ {\begin{array}{*{20}{c}} {1 - 1}&{2( - 1)} \\\ {2( - 1)}&{1 - 1} \end{array}} \right]
Hence, on simplifying
$\Rightarrow$ A = \left[ {\begin{array}{*{20}{c}} 0&{( - 1)} \\\ {( - 1)}&0 \end{array}} \right]
Hence, from both the above contents we can see that the reflection of line along the line given $x + y = 0$ is

0&{ - 1} \\\ { - 1}&0 \end{array}} \right]$$ Hence, the required matrix of the transformation reflection in the line $$x + y = 0$$ is $$ = \left[ {\begin{array}{*{20}{c}} 0&{ - 1} \\\ { - 1}&0 \end{array}} \right]$$ **Hence, option (d) is our required correct answer.** **Note:** When you reflect a point across the line $$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)$$ . Hence, use graph methods also in order to take the reflection of the given quantities. And also remember the concept of Matrix as $$A = \dfrac{1}{{1 + {m^2}}}\left[ {\begin{array}{*{20}{c}} {1 - {m^2}}&{2m} \\\ {2m}&{{m^2} - 1} \end{array}} \right]$$ . Hence, substitute the value correctly. Here, m is the slope of the line.