2
$\begingroup$

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 $y=-\sqrt{3}x$.

The question asks: What is

$A^{1001}$

$B^{1001}$

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!

$\endgroup$
  • 1
    $\begingroup$ I agree with your three answers. $\endgroup$ – José Carlos Santos Sep 6 '18 at 20:27
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.