How many solutions exist for a matrix equation $A^2=I$?

Question

Let $A$ be a square matrix of order three or two, and $I$ be a unit matrix. How many solutions are possible for the equation $$A^2=I$$? In case the solutions are infinite, or very large, how do I check, without actual multiplication, if a given matrix is a solution to the given equation. In the general case, what about a factorisable matrix polynomial equation?

Answer 1

A matrix $A$ will satisfy $p(A)=0$ for a polynomial $p$ if and only if the minimal polynomial of $A$ divides $p$.

In your case, we have $p(x) = x^2 - 1$. So, the minimal polynomial of $A$ must be $x^2-1,x-1,$ or $x+1$. In any case, we can say

$A^2 = I$ if and only if $A$ is diagonalizable and has eigenvalues $1$ or $-1$

In the size $2$ case, we have $A=I,A=-I$, or $$ A = S\pmatrix{1&0\\0&-1}S^{-1} $$ to check whether $A$ is one of these last matrices, we can verify that trace$(A) = 0$ and $\det(A) = -1$.

Answer 2

Notice in general (unless the field is finite) there could be uncountably infinite many solutions even for $2\times 2 $ matrices, e.g. $$\forall c\in\mathbb R:A= \left( \begin{array}{ccc} 1 & 0 \\ c & -1 \\ \end{array} \right) $$