Question

Hey Super Sleuths! I have a quick question that I'd like to confirm... Find all 2 x 2 matrices such that A^2 = I_2 (the identity).
I have found 6, is this all of them?
Thanks!

Answer 1

\[\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}, \begin{bmatrix} a & 1 + a \\ 1 - a & -a \end{bmatrix}, \begin{bmatrix} a & 1 - a \\ 1 + a & -a \end{bmatrix}\]

Answer 2

define\[A=\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]\]\[A^2=\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]=\left[\begin{matrix}a^2+bc & ab+bd \\ ac+dc & bc+d^2\end{matrix}\right]=I_2=\left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right]\]so we have\[a^2+bc=1\]\[b(a+d)=0\]\[c(a+d)=0\]\[bc+d^2=1\]