Question

Let \(A\) be a \(3\times 3\) real matrix such that given any column vector \(x\in \mathbb{R}^3\), the column vector \(Ax\) is the reflection of \(x\) about the plane

Hint:

For reflection about a plane through the origin in \(\mathbb{R}^3\), eigenvalues are \(1,1,-1\), so the trace is \(1\).

Answer 1

Correct Answer: 1

Step 1: Find the plane.
The points $(a,b,-a-b)$ satisfy $x+y+z=0$, so the mirror plane is $x+y+z=0$, passing through the origin.

Step 2: Reflection eigenvalues.
A reflection in a plane keeps vectors lying in the plane fixed, so those give eigenvalue $1$.

Step 3: The plane is two dimensional.
The plane $x+y+z=0$ has dimension $2$, so eigenvalue $1$ shows up twice.

Step 4: The normal flips.
The normal direction $(1,1,1)$ is sent to its opposite, giving eigenvalue $-1$.

Step 5: Use the trace.
The sum of the diagonal entries equals the sum of eigenvalues, $1+1-1=1$.
\[ \boxed{1.0} \]