TITLE: Is $A^2=I \implies A=\pm I$ necessarily true?
QUESTION [5 upvotes]: $A$ is $n\times n$ matrix.
How to prove whether  it is true or false $$A^2=I \implies A=\pm I$$
I was trying on $2\times 2$ case...multiplying general entries and then equating them to the identity requirements...but it is not a proof...

REPLY [3 votes]: Any time that you get back to where you started by doing an operation twice, you have the situation $A^2=I$. Such operations are called involutions. So, in the plane, reflection about any axis that passes through the origin will have this property. Turning your shirt inside-out is probably not a linear operation, so it won’t have a matrix, but it is an involution. These operations are everywhere, even though you can’t unring a bell.