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 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

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)$$.