Question

Show that if a 3 x 3 matrix A represents the reflection about a plane, then A is similar to the matrix $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$.

Short answer

If a 3 x 3 matrix A represents the reflection about a plane, then A is similar to the matrix $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$.

Step-by-step solution

  Considering an equation of a plane

Let us consider$x_1-2x_2+2x_3=0$is an equation of a plane.

Also let us consider the basis of the plane consists of two vectors parallel to the plane and one vector perpendicular to the plane.

i.e.$I=\left\{u⃗,v⃗,w⃗\right\}$is the basis of the plane such thatlocalid="1659354468607" $\stackrel{\rightharpoonup }{U},\stackrel{\rightharpoonup }{V}$are parallel to the plane and is perpendicular to the plane. Then we have $\stackrel{\rightharpoonup }{w}=\stackrel{\rightharpoonup }{U}\times \stackrel{\rightharpoonup }{V}$.

 Step 2: Finding the reflection matrix A about the plane x1-2x2+2x3=0

Since$I=\left\{u⃗,v⃗,w⃗\right\}$ is the basis of the plane such thatrole="math" localid="1659354529564" $\stackrel{\rightharpoonup }{U},\stackrel{\rightharpoonup }{V}$ are parallel to the plane androle="math" localid="1659354504549" $\stackrel{\rightharpoonup }{w}$ is perpendicular to the plane, then the linear transformation of the vectors $\stackrel{\rightharpoonup }{U},\stackrel{\rightharpoonup }{V}$and$\stackrel{\rightharpoonup }{w}$ is given by

role="math" localid="1659354641352" $$\begin{gathered} T\left(\stackrel{\rightharpoonup }{u}\right)=\stackrel{\rightharpoonup }{u} \\ T(v⃗)=\stackrel{\rightharpoonup }{v} \\ T(\stackrel{\rightharpoonup }{w}⃗)=-\stackrel{\rightharpoonup }{w} \end{gathered}$$

Therefore the matrix $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$which is clearly similar to the given matrix .

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$

  Final answer

If a 3 x 3 matrix A represents the reflection about a plane, then A is similar to the matrix $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$