Products of orthogonal projection and reflection matrices

I am asking about a 3 part problem that I was marked for only 3/10 points.

Let the matrix $A$ represent the orthogonal projection onto the line $y=-\sqrt{3}x$, and let matrix $B$ represent reflection over the line $y=-\sqrt{3}x$.

$A^{1001}$

$B^{1001}$

and $AB$

I thought that $A^{1001}=A$ because an orthogonal projection that is performed repeatedly is still an orthogonal projection. Is this correct? (the grader doesn't specify which parts of the problem I got wrong)

I also answered that $B^{1001}=B$ because a reflection performed an odd number of times, like 1, 3, or 5 times, is in fact the same reflection, whereas performing it an even number of times would return to the original position. i.e. $B^{1000}=I$.

For the last part, I said that $AB=A$ because $AB$ represents a reflection and a subsequent projection, which would yield the same result as a projection straight away.

Let me know what I'm missing here or perhaps what the grader is missing. Thank you!

• I agree with your three answers. – José Carlos Santos Sep 6 '18 at 20:27

Your reasoning is right. It is not hard to see that $B=2A-I$. Then $$AB=A(2A-I)=2A-A=A.$$ The only thing I can imagine is that the marker expected more detail, but I cannot tell without context.