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.
