I need to find all 2x2 matrices A,
(a b)
(c d)
such that A^2=I2, the 2x2 identity matrix.
Just to flesh out what tah said:
you have $\displaystyle \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
now, compute the matrix product on the left hand side, then equate corresponding components to get your 4 equations that tah spoke of. solve this system for your solution