# Powers of matrices equal to the identity matrix

by Bipolarity
Tags: equal, identity, matrices, matrix, powers
 P: 774 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
 Engineering Sci Advisor Thanks P: 6,067 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.
 P: 774 I see. Thanks much $\aleph_0$ BiP
P: 144

## Powers of matrices equal to the identity matrix

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.

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