Question

Find 8 3x3 matricies such that
A^2 = I

Answer 1

What I have so far:

Let A = \[\left[\begin{matrix}a & b & c\\ d & e & f \\ g & h & i\end{matrix}\right]\]
Then A^2 gives us the following system of equations:
1 = a^2 + bd + ci = e^2 + bd + fh = i^2 + cg + fh
0 = ab + be + ch = ac + bf + ci = ad + de + fg = cd + ef + fi = ag + dh + ci = bg + eh + hi

We can set a,e,i to 1 and -1 (since -1^2 = 1)
\[\left[\begin{matrix}1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]\]
\[\left[\begin{matrix}-1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & -1\end{matrix}\right]\]

now I'm stuck finding 6 more

Answer 2

The textbook's hint is "Look for solutions in which all entries off the main diagonal are zero"

Answer 3

Wait.....I can just do something like:
(a,e,i)
then all of these work:
(1,1,-1)
(1,-1,1)
(-1,1,1)
(1,-1,-1)
(-1,-1,1)
(-1,1,-1)

And that's 6 more solutions lol.