# About finding $2\times 2$ matrices that are their own inverses

They ask me to find all invertible matrices $A$ of the form: $\begin{bmatrix}a & b\\ c&d \end{bmatrix}$ and satisfying $A=A^{-1}$ and $A^t=A^{-1}$. I find that rather complex; does it have anything to do with orthogonality? Not sure. Any help please??

#### Solutions Collecting From Web of "About finding $2\times 2$ matrices that are their own inverses"

Hint: Given a 2×2 matrix, do you know how to compute its inverse? If not then figure out how to do that. Now that you do, take an arbitrary invertible 2×2 matrix (that is call its entries $a,b,c,d$), compute its inverse and solve the equations $A=A^{-1}$ and $A^t=A^{-1}$.

Matrices of real entries such that $AA^t=I$ are called unitary matrices and the columns of such matrices must be orthogonal.

and you want to find unitary matrices such that $A^t=A$ which are called symmetric matrices, so the answer will be all symmetric unitary matrices.