Periodic Matrix – Definition and Example

Here you will learn what is periodic matrix with examples.

Let’s begin –

Periodic Matrix

A square matrix which satisfies the relation $$A^{k+1}$$ = A for some positive integer k, is called a periodic matrix.

The period of the matrix is the least value of k for which $$A^{k+1}$$ = A holds true.

Note that the period of idempotent matrix is 1.

Example : Find the period of the matrix A = $$\begin{bmatrix} 1 & -2 & -6 \\ -3 & 2 & 9 \\ 2 & 0 & -3 \end{bmatrix}$$.

Solution : We have,

A = $$\begin{bmatrix} 1 & -2 & -6 \\ -3 & 2 & 9 \\ 2 & 0 & -3 \end{bmatrix}$$.

Now, $$A^2$$ = A.A

$$\implies$$ $$A^2$$ = $$\begin{bmatrix} 1 & -2 & -6 \\ -3 & 2 & 9 \\ 2 & 0 & 3 \end{bmatrix}$$ $$\times$$ $$\begin{bmatrix} 1 & -2 & -6 \\ -3 & 2 & 9 \\ 2 & 0 & 3 \end{bmatrix}$$

= $$\begin{bmatrix} 5 & -6 & -6 \\ 9 & 10 & 9 \\ -4 & -4 & -3 \end{bmatrix}$$.

Now, $$A^3$$ = $$A^2$$.A

$$\implies$$ $$A^3$$ = $$\begin{bmatrix} 5 & -6 & -6 \\ 9 & 10 & 9 \\ -4 & -4 & -3 \end{bmatrix}$$ $$\times$$ $$\begin{bmatrix} 1 & -2 & -6 \\ -3 & 2 & 9 \\ 2 & 0 & -3 \end{bmatrix}$$

= $$\begin{bmatrix} 1 & -2 & -6 \\ -3 & 2 & 9 \\ 2 & 0 & -3 \end{bmatrix}$$ = A

Hence, $$A^3$$ = A. comparing it with the equation $$A^{k+1}$$ = A gives k = 2.

So, Period of the given matrix is 2.