Powers of matrices equal to the identity matrix

[Bipolarity]

I am curious about under what conditions the powers of a square matrix can equal the identity matrix.

Suppose that A is a square matrix so that $A^{2} = I$

At first I conjectured that A is also an identity matrix, but I found a counterexample to this.
I noticed that the counterexample was an elementary matrix.

So then I conjectured that A is an elementary matrix. Is this true? Can I prove this? What about for general powers of A?

BiP

 

[AlephZero]

As a simple example think about 2x2 matrices.

If $\displaystyle A = \begin{bmatrix}a & b \\ c & d \end{bmatrix}$, then

$\displaystyle A^2 = \begin{bmatrix}a^2 + bc & b(a+d) \\ c(a+d) & d^2 + bc \end{bmatrix} = I$.

From the off-diagonal terms, $b(a+d) = 0$ and $c(a+d) = 0$.
Taking $b = c = 0$ isn't going to lead to anywhere interesting, so let's see what happens if $d = -a$.
From the diagonal terms, $a^2 + bc = 1$.

You can satisfy that with matrices that are not elementary, for example
$\displaystyle A = \begin{bmatrix} 2 & 3 \\ -1 & -2 \end{bmatrix}$.

In fact the condition $a^2 + bc = 1$ here is the same as $|\det A| = 1$, which isn't a complete coincidence - but things are not so simple for bigger matrices.

 

[Bipolarity]

I see. Thanks much $\aleph_0$

BiP

 

[Norwegian]

A solution to Aⁿ=I is obviously attained if A is a suitable diagonal or rotation matrix, and also for all similar matrices PAP^-1, where P is invertible.