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.

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