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.

 

[blue_raver22]

olarScientist:

Hello there! It is great to see your curiosity in exploring the powers of matrices equal to the identity matrix. This is an interesting topic in linear algebra and has many applications in various fields of science and engineering.

To answer your question, it is not necessary for a matrix A to be an identity matrix in order for its powers to equal the identity matrix. As you have discovered, there are counterexamples to this conjecture, such as elementary matrices. These are matrices that can be obtained by performing a single elementary row operation on the identity matrix.

As for your conjecture that A must be an elementary matrix, this is also not necessarily true. There are other types of matrices that can satisfy the condition A^{2} = I, such as diagonal matrices with entries of 1 or -1 along the main diagonal.

In general, we can say that if a square matrix A has the property that A^{n} = I for some positive integer n, then A is called an nth root of the identity matrix. However, there is no guarantee that A will be an elementary matrix or have any specific structure.

In order to prove that A is an elementary matrix, you would need to show that it can be obtained from the identity matrix by performing a single elementary row operation. This may not always be the case for all powers of A, making it difficult to prove your conjecture.

I hope this helps answer your questions and encourages you to continue exploring the fascinating world of linear algebra. Keep asking questions and seeking answers, as that is the essence of science. Best of luck in your research!