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 [itex] A^{2} = I [/itex]

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 [itex]\aleph_0[/itex]

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.